Lens array sheet surface light source, and transmission type display device

ABSTRACT

A lens array sheet according to the present invention, comprising a transparent substrate, a lens array having lens elements that are one-dimensionally or two-dimensionally formed on the front surface of the transparent substrate, and a cluster having a large number of cluster members randomly formed in a prism shape on the rear surface of the transparent substrate, each of the length, the width, and the height of each of the cluster members being in the range from the wave length of source light to 500 μm. Thus, a lens array sheet that effectively uses light energy of the light source, maintains the light condensing effect, prevents the luminance from deteriorating, homogeneously distributes the luminance on the light emitting surface, prevents equal-thickness interference fringes and wasteful light dispersion to out of the angular range of visual angle can be provided. In addition, a surface light source having the lens array sheet is provided. Moreover, a bright transmission type display device having the surface light source is provided.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a lens array sheet that has ahomogeneous lighting characteristic as a lighting means for use with atransmission type liquid crystal display device, a back light source fora transmission type display device for an advertisement board. Inaddition, the present invention relates to a surface light source and atransmission type display device with the lens array sheet.

2. Description of the Related Art

In recent years, requirements for low weight and low power consumptionhave been made for transmission type liquid crystal devices. Varioussurface light sources that effectively use light emitted from lightsources and guide the resultant light only to a necessary andsatisfactory direction have been proposed.

In these related art references, a light source is disposed on a sidesurface of an optical conductor composed of a plate of, for example, atransparent acrylic resin. The light entered from the side surface intothe optical conductor is reflected on a reflection layer on the rearsurface of the optical conductor. The light is emitted from the lightemitting surface that is the upper surface of the optical conductor(light guide). At this point, to homogenize the light, a diffusion sheetis disposed on the upper surface of the optical conductor.Alternatively, to condense the light as emitted light only withinpredetermined angular range, a lens array sheet that operates as a lensis disposed as a surface light source. The surface light source of whichthe light source is disposed on the side surface of the opticalconductor is referred to as an edge light type surface light source.

Although a box-type planar light source of which a light source isdisposed immediately below a diffusion sheet or a lens array sheet isknown, since the thickness thereof increases, the application thereof islimited.

As described above, various methods for effectively using light emittedfrom light sources without loss have been proposed. As an example, alens array sheet for condensing light as emitted light within apredetermined angular range is known. As shown in FIG. 24, the linearlens array sheet is composed of a large number of triangular prisms aslens elements that are one-dimensionally arranged so that their edgelines are arranged in parallel. In addition, a two-layer type linearlens array sheet has been proposed so as to condensed more light andimprove the luminance.

For example, two-layer type linear lens array sheets of which lineartriangular prisms are arrayed as lens elements have been disclosed inJapanese Patent Laid-Open Publication Nos. 5-203950, 5-313156, and5-313164.

However, although the two-layer lens array sheet has an advantage of animprovement of luminance due to the light condensing effect, it also hasthe following disadvantage. In the lens array sheet of which the lenselements are arrayed on the front surface and of which the rear surfaceis flat, the rear surface of the upper lens array sheet microscopicallycontacts the vertex portions of the lens elements of the lower lensarray sheet. Thus, the optically transparent contact portion accordswith the vertex portions of the lower lens elements. Consequently, thelens vertex portions become visible. When the lens elements aretriangular prisms, the vertex portions are shaped as edge lines. As aresult, many lines are visible.

Due to small differences of the distances of lens elements of the twolens array sheets, Newton rings that are equal-thickness interferencefringes as a concentric circle pattern or a concentric ellipse patternmay occur on the entire surface of the surface light source.

To prevent such a problem, the rear surface of the lens array sheet ismatted so as to form small concave and convex portions (hereinafterreferred to as a cluster). Thus, the lens array sheets are preventedfrom contacting. This method has been disclosed in Japanese PatentLaid-Open Publication No. 7-151909.

However, when the rear surface of the lens array sheet is matted, lightis diffuse-reflected on the matted surface. Thus, the matted surfaceoperates as a light diffusion sheet. Consequently, the function of thelens array sheet, which condenses light in a desired diffusion anglewithin a desired diffusion angle is remarkably deteriorated. Thus, theluminance is remarkably decreased. In addition, since the height of eachportion of the cluster of the matted surface is not completelyhomogeneous, there are small differences in the distances of the lenselements of the two lens array sheets (hereinafter, each portion of thecluster is referred to as a cluster member). Thus, the equal-thicknessinterference fringes tend to take place.

On the other hand, in the structure of which only one lens array sheetis used, in the case that the rear surface of the lens array sheet issmooth, when the lens array sheet is disposed on the light emittingsurface of an optical conductor of an edge light type surface lightsource, since the lens array sheet contacts the light emitting surfaceof the optical conductor, they are optically unified. Thus, the emittedlight of the light source cannot be homogeneously total-reflected on thefront surface of the optical conductor. Even if spacers are disposed atfour corners of the optical conductor or the lens array sheet so as tohave a space between the optical conductor and the lens array sheet,since the lens array sheet is bent and deformed, small differences indistances of the lens elements of the lens array sheet and the opticalconductor cause the equal-thickness interference fringes to take place.To prevent this problem, a structure of which a cluster with a height ofthe wave length of a source light or larger is formed has been disclosedin for example Japanese Patent Laid-Open Publication Nos. 5-323319 and6-324205.

However, the cluster is formed as an optically-homogeneous diffusionpattern such as a sand-face pattern or a pear-face pattern. Thus, partof light emitted from the optical conductor diffuses out of the angularrange of visual field. Consequently, the light condensing effect of thelens array sheet deteriorates and thereby wasting the energy of thelight of the light source and deteriorating the luminance.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a lens array sheet thatcan solve the above-described problems, effectively use light energy ofthe light source, maintain the light condensing effect, prevent theluminance from deteriorating, homogeneously distribute the luminance onthe light emitting surface, prevent equal-thickness interference fringesand wasteful light dispersion to out of the angle of visual angle fromtaking place. Another object of the present invention is to provide asurface light source having the lens array sheet. A further other objectof the present invention is to provide a light transmission type displaydevice having the surface light source.

To accomplish the objects, a lens array sheet according to the firstaspect of the present invention comprises a transparent substrate, alens array having lens elements that are one-dimensionally ortwo-dimensionally formed on the front surface of the transparentsubstrate, and a cluster having a large number of cluster membersrandomly formed in a prism shape on the rear surface of the transparentsubstrate, each of the length, the width, and the height of each of thecluster members being in the range from the wave length of source lightto 500 μm.

In the lens array sheet, each of the cluster members may be formed in arectangular parallelepiped shape.

In the lens array sheet, the line of intersection of a horizontalsurface of the lens array sheet and a surface that composes the lenselements may be not in parallel with the line of intersection of thehorizontal surface and a side surface of each of the rectangularparallelepiped shape of the cluster members.

A surface light source according to the first aspect of the presentinvention comprises an optical conductor composed of at least atransparent flat plate, a light source unit disposed adjacent to atleast one of side edge surfaces of the optical conductor, a lightreflection layer formed on the rear surface of the optical conductor,and one or two lens array sheets of the first aspect of the presentinvention and disposed on a light emitting surface of the front surfaceof the optical conductor so that the lens array sheets face the frontsurface of the optical conductor.

In the edge light type surface light source, the lens array sheet may becomposed of two lens array sheet members that are layered, the clusterof the lower lens array sheet member facing the front surface of theoptical conductor.

The edge light type surface light source may comprise a light diffusionsheet formed on the light emitting surface that is the front surface ofthe optical conductor and having concave and convex portions on thefront and rear surfaces, the height of the concave and convex portionsbeing homogeneous to or greater than the wave length of source light, arear-surface flat lens array sheet having a lens array with lenselements that are one-dimensionally or two-dimensionally formed on thefront surface of the transparent substrate, the rear surface of therear-surface flat lens array sheet facing the front surface of theoptical conductor, and a lens array sheet of the first aspect of thepresent invention, wherein the light diffusion sheet, the rear-surfaceflat lens array sheet, and the lens array sheet of the present inventionare layered in the order.

The transparent type display device of the first aspect of the presentinvention comprises a surface light source of the first aspect of thepresent invention, the surface light source being used as a back lightsource for the transmission type display device.

Thus, according to the transmission type display device of the firstaspect of the present invention, since the cluster is formed on the rearsurface, when two lens array sheets are layered or a lens array sheet isdisposed as a surface light source on the light emitting surface of theoptical conductor, the rear surface of the lens array sheet can beprevented from being contacted, thereby suppressing the equal-thicknessinterference fringes from taking place. When the lens array sheet isdisposed on the optical conductor in such a manner that the rear surfaceof the lens array sheet (on which the cluster is formed) faces theoptical conductor, since the light distribution is not affected by thetotal reflection on the front surface of the optical conductor, light ishomogeneously emitted with a homogeneous luminance distribution on theentire surface of the light emitting surface. In other words, the amountof light that is emitted out of the angular range of visual field isreduced and thereby the decrease of the luminance within angular rangesof visible field can be minimized in comparison with the conventionalcontact preventing method using the mat process.

In addition, since the cluster members that compose the cluster arerandomly formed, the moire fringes due to the interference of lens arrayor pixel array of the liquid crystal display device and the clustermembers can be prevented.

When the cluster members are formed in a rectangular parallelepipedshape, the transmission type display device can be easily fabricated. Inaddition, when the relation between the side surface of the rectangularparallelepiped cluster and the surface of the lens elements of the lensarray is designated in a predetermined manner, the moire fringes due tothe lens array can be prevented.

According to the edge light type surface light source of the firstaspect of the present invention, since the lens array sheet does notcontact the light emitting surface of the optical conductor, the lightemitted from the light source can be widely and homogeneouslydistributed in the optical conductor. Thus, the luminance distributionof the light emitted from the optical conductor can be homogenized onthe light emitting surface. In addition, the light energy can beeffectively used and thereby be bright. Moreover, the light diffusiondot pattern which is formed in the rear surface of the optical conductorcan become invisible. The amount of light that is emitted in thevicinity of the normal direction of the light emitting surface is large.In addition, the amount of light that is emitted in other than thenormal direction can be reduced in comparison with the homogeneousdiffusion sheet.

According to the transmission type display device of the first aspect ofthe present invention, the light emitted from the display surface isbright on the entire surface regardless of the angular range of visualfield within the predetermined angular range.

To accomplish the objects, a lens array sheet according to the secondaspect of the present invention comprises a transparent substrate, alens array having lens elements that are one-dimensionally ortwo-dimensionally formed on the front surface of the transparentsubstrate, and a cluster having a large number of cluster membersrandomly formed in a prism shape on the rear surface of the transparentsubstrate, each of the length, the width, and the height of each of thecluster members being in the range from the wave length of source lightto 500 μm.

The random two-dimensional distribution is a distribution of which theposition of each lattice point of the two-dimensional periodic array israndomly moved and reallocated.

In lens array sheet, each of the cluster members may be formed in arectangular parallelepiped shape.

In the lens array sheet, the line of intersection of a horizontalsurface of the lens array sheet and a surface that composes the lenselements may be not in parallel with the line of intersection of thehorizontal surface and a side surface of each of the rectangularparallelepiped shape of the cluster members.

A surface light source according to the second aspect of the presentinvention comprises an optical conductor composed of at least atransparent flat plate, a light source unit disposed adjacent to atleast one of side edge surfaces of the optical conductor, a lightreflection layer formed on the rear surface of the optical conductor,and one or two lens array sheets of the second aspect of the presentinvention and disposed on a light emitting surface of the front surfaceof the optical conductor so that the lens array sheets face the frontsurface of the optical conductor.

In the edge light type surface light source, the lens array sheet may becomposed of two lens array sheet members that are layered, the clusterof the lower lens array sheet member facing the front surface of theoptical conductor.

The edge light type surface light source may comprise a light diffusionsheet formed on the light emitting surface that is the front surface ofthe optical conductor and having concave and convex portions on thefront and rear surfaces, the height of the concave and convex portionsbeing homogeneous to or greater than the wave length of source light, arear-surface flat lens array sheet having a lens array with lenselements that are one-dimensionally or two-dimensionally formed on thefront surface of the transparent substrate, the rear surface of therear-surface flat lens array sheet facing the front surface of theoptical conductor, and a lens array sheet of the second aspect of thepresent invention, wherein the light diffusion sheet, the rear-surfaceflat lens array sheet, and the lens array sheet are layered in thatorder.

A transparent type display device according to the second aspect of thepresent invention comprises a surface light source of the second aspectof the present invention.

Thus, according to the transmission type display device of the secondaspect of the present invention, since the cluster is formed on the rearsurface, when two lens array sheets are layered or a lens array sheet isdisposed as a surface light source on the light emitting surface of theoptical conductor, the rear surface of the lens array sheet can beprevented from being contacted, thereby suppressing the equal-thicknessinterference fringes from taking place. When the lens array sheet isdisposed on the optical conductor in such a manner that the rear surfaceof the lens array sheet (on which the cluster is formed) faces theoptical conductor, since the light distribution is not affected by thetotal reflection on the front surface of the optical conductor, light ishomogeneously emitted with a homogeneous luminance distribution on theentire surface of the light emitting surface. In other words, the amountof light that is emitted out of the angular range of visual field isreduced and thereby the decrease of the luminance can be minimized incomparison with the conventional contact preventing method using the matprocess. In particular, since the cluster members are randomlydistributed with a homogeneous surface density of the number of clustermembers, uneven luminance does not take place.

In addition, since the cluster members that compose the cluster arerandomly formed by the predetermined randomizing method, the homogeneousluminance distribution without an uneven distribution of the density ofthe cluster member regardless of the number of the cluster members canbe accomplished and thereby the moire fringes due to the interference oflens array and pixel array of the liquid crystal display device can beprevented.

When the cluster members are formed in a rectangular parallelepipedshape, the transmission type display device can be easily fabricated. Inaddition, when the relation between the side surface of the rectangularparallelepiped cluster and the surface of the lens elements of the lensarray is designated in a predetermined manner, the moire fringes due tothe lens array can be prevented.

According to the edge light type surface light source of the secondaspect of the present invention, since the lens array sheet does notcontact the light emitting surface of the optical conductor, the lightemitted from the light source can be widely and homogeneouslydistributed in the optical conductor. Thus, the luminance distributionof the light emitted from the optical conductor can be homogenized onthe light emitting surface. In addition, the light energy can beeffectively used and thereby be bright. Moreover, the light diffusiondot pattern can become invisible. The amount of light that is emitted inthe vicinity of the normal direction of the light emitting surface islarge. In addition, the amount of light that is emitted in other thanthe normal direction can be reduced in comparison with the homogeneousdiffusion sheet.

According to the transmission type display device of the second aspectof the present invention, the light emitted from the display surface isbright on the entire surface regardless of the angular range of visualfield.

To accomplish the objects, a lens array sheet according to the thirdaspect of the present invention comprises a transparent substrate, alens array having lens elements that are one-dimensionally ortwo-dimensionally formed on the front surface of the transparentsubstrate, and a cluster having a large number of cluster membersrandomly formed in a prism shape on the rear surface of the transparentsubstrate, wherein each of the cluster members is composed of clustermember elements formed in a prism shape or a truncated prismoid shape,each of the length of the minimum diagonal line and the height of thetop and bottom of each of the cluster member elements being homogeneousto or greater than the wave length of the source light, each of thelength of the maximum diagonal line and the height thereof beinghomogeneous to or smaller then 500 pm, the cluster member elements beingallocated to structural elements of a percolation cluster in atwo-dimensional lattice with a critical percolation concentration Pc orsmaller, adjacent cluster member elements being fused.

In the lens array sheet, lattice points of the two-dimensional latticemay be allocated to the cluster member elements with an occupyingprobability P that may be smaller than the critical percolationconcentration Pc, the cluster members being composed by fusing theadjacent cluster member elements.

In the lens array sheet, the two-dimensional lattice may be a squarelattice, the lattice member elements allocated to the lattice pointsbeing formed in a rectangular parallelepiped shape.

In the lens array sheet, the line of intersection of the horizontalsurface of the lens array sheet and the surface composing each of thelens elements may be not in parallel with the line of intersection ofthe horizontal surface and a side surface of each of the clustermembers.

A surface light source according to the third aspect of the presentinvention comprises an optical conductor composed of at least atransparent flat plate, a light source unit disposed adjacent to atleast one of side edge surfaces of the optical conductor, a lightreflection layer formed on the rear surface of the optical conductor,and one or two lens array sheets of the third aspect of the presentinvention and disposed on a light emitting surface of the front surfaceof the optical conductor so that the lens array sheets face the frontsurface of the optical conductor.

In the edge light type surface light source, the lens array sheet may becomposed of two lens array sheet members that are layered, the clusterof the lower lens array sheet member facing the front surface of theoptical conductor.

The edge light type surface light source may comprise a light diffusionsheet formed on the light emitting surface that is the front surface ofthe optical conductor and having concave and convex portions on thefront and rear surfaces, the height of the concave and convex portionsbeing homogeneous to or greater than the wave length of source light, arear-surface flat lens array sheet having a lens array with lenselements that are one-dimensionally or two-dimensionally formed on thefront surface of the transparent substrate, the rear surface of therear-surface flat lens array sheet facing the front surface of theoptical conductor, and a lens array sheet of the third aspect of thepresent invention, wherein the light diffusion sheet, the rear-surfaceflat lens array sheet, and the lens array sheet are layered in theorder.

A transparent type display device according to the third aspect of thepresent invention comprises a surface light source of the third aspectof the present invention, the surface light source being used as a backlight source for the transmission type display device.

Thus, according to the transmission type display device of the thirdaspect of the present invention, since the cluster is formed on the rearsurface, when two lens array sheets are layered or a lens array sheet isdisposed as a surface light source on the light emitting surface of theoptical conductor, the rear surface of the lens array sheet can beprevented from being contacted, thereby suppressing the equal-thicknessinterference fringes from taking place. When the lens array sheet isdisposed on the optical conductor in such a manner that the rear surfaceof the lens array sheet (on which the cluster is formed) faces theoptical conductor, since the light distribution is not affected by thetotal reflection on the front surface of the optical conductor, light ishomogeneously emitted with a homogeneous luminance distribution on theentire surface of the light emitting surface. In other words, the amountof light that is absorbed or emitted out of the angular range of visualfield is reduced and thereby the decrease of the luminance can beminimized in comparison with the conventional contact preventing methodusing the mat process.

In particular, the shape of the cluster is fractal and randomly formed.Thus, although the average rotating radius is large, non-clusterportions are randomly formed in the radius. Consequently, the cluster isinvisible and uneven luminance hardly takes place.

In addition, since the cluster members that compose the cluster arerandomly shaped and formed by the predetermined randomizing methodcorresponding to the theory of percolation, the homogeneous luminancedistribution without an uneven distribution of the density of thecluster member regardless of the number of the cluster members can beaccomplished and thereby the moire fringes due to the interference oflens array and pixel array of the liquid crystal display device can beprevented.

When the cluster members are formed in a rectangular parallelepipedshape, the transmission type display device can be easily fabricated. Inaddition, when the relation between the side surface of the rectangularparallelepiped cluster and the surface of the lens elements of the lensarray is designated in a predetermined manner, the moire fringes due tothe lens array can be prevented.

According to the edge light type surface light source of the thirdaspect of the present invention, since the lens array sheet does notcontact the light emitting surface of the optical conductor, the lightemitted from the light source can be widely and homogeneouslydistributed in the optical conductor. Thus, the luminance distributionof the light emitted from the optical conductor can be homogenized onthe light emitting surface. In addition, the light energy can beeffectively used and thereby be bright. Moreover, the light diffusiondot pattern can become invisible. The amount of light that is emitted inthe vicinity of the normal direction of the light emitting surface islarge. In addition, the amount of light that is emitted in other thanthe normal direction can be reduced in comparison with the homogeneousdiffusion sheet.

According to the transmission type display device of the third aspect ofthe present invention, the light emitted from the display surface isbright on the entire surface regardless of the angular range of visualfield.

These and other objects, features and advantages of the presentinvention will become more apparent in light of the following detaileddescription of best mode embodiments thereof, as illustrated in theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view showing a lens array sheet according to anembodiment of a first aspect and a second aspect of the presentinvention;

FIG. 2 is a perspective view showing the shape of a cluster member ofthe lens array sheet according to the embodiment of the first aspect andthe second aspect of the present invention;

FIG. 3 is an enlarged perspective view for explaining cluster membersformed on the lens array sheet;

FIGS. 4A and 4B are schematic diagrams for explaining that the sidesurfaces of the cluster members are not in parallel with the structuralsurface of the lens array;

FIGS. 5A, 5B, and 5C are schematic diagrams for explaining a process forrandomly forming the cluster members;

FIGS. 6A and 6B are schematic diagrams for explaining a process forforming cluster members that overlap;

FIGS. 7A, 7B, 7C, 7D, and 7E are schematic diagrams for explaining atwo-dimensional lattice and a randomized cluster corresponding to thesecond aspect of the present invention;

FIG. 8 is a perspective view showing a lens array sheet according to athird aspect of the present invention;

FIG. 9 is a perspective view showing an example of a prism shape clustermember of the lens array sheet according to the third aspect of thepresent invention;

FIG. 10 is a perspective view for explaining a perpendicularparallelepiped shape as a prism that composes a cluster member;

FIGS. 11A and 11B are schematic diagrams for explaining that the sidesurfaces of the cluster members is not in parallel with the structuralsurface of the lens array.

FIGS. 12A and 12B are schematic diagrams for explaining cluster membersformed corresponding to the percolation of a two-dimensional lattice,FIG. 12A showing cluster members occupied in squares of the latticepoints of the two-dimensional lattice with a predetermined occupyingprobability, FIG. 12B showing a cluster generated by connectingvertically and horizontally adjacent squares shown in FIG. 12A;

FIGS. 13A, 13B, and 13C are schematic diagrams for explaining the fusingof the cluster members;

FIGS. 14A and 14B are schematic diagrams showing examples oftwo-dimensional lattices, FIG. 14A showing a basket-weave-shape lattice,FIG. 14B showing a hexagonal lattice;

FIG. 15 is a schematic diagram showing a cluster in the case that anoccupying probability P exceeds a critical percolation concentration Pc;

FIG. 16 is a schematic diagram showing a first example of the clustercorresponding to the occupying probability P (No. 1);

FIG. 17 is a schematic diagram showing another example of the clustercorresponding to the occupying probability P (No. 2);

FIG. 18 is a schematic diagram showing another example of the clustercorresponding to the occupying probability P (No. 3);

FIG. 19 is a schematic diagram showing another example of the clustercorresponding to the occupying probability P (No. 4);

FIG. 20 is a schematic diagram showing another example of the clustercorresponding to the occupying probability P (No. 5);

FIG. 21 is a vertical sectional view showing a layer structure of aone-layer type lens array sheet according to an embodiment of the firstto third aspects of the present invention;

FIG. 22 is a vertical sectional view showing a layer structure of atwo-layer type lens array sheet according to an embodiment of the firstto third aspects of the present invention;

FIGS. 23A and 23B are vertical sectional views showing layer structuresof three-layer type lens array sheets according to embodiments of thefirst to third aspects of the present invention;

FIG. 24 is a perspective view showing an example (triangular prism lens)of the lens array of the lens array sheet according to the first tothird aspects of the present invention;

FIG. 25 is a perspective view showing another example (ellipticalcylinder lens) of the lens array of the lens array sheet according tothe first to third aspects of the present invention;

FIG. 26 is a perspective view showing another example (concave lens) ofthe lens array of the lens array sheet according to the first to thirdaspects of the present invention;

FIG. 27 is a perspective view showing another example (fly-eye lens) ofthe lens array of the lens array sheet according to the first to thirdaspects of the present invention;

FIG. 28 is a perspective view showing another example (pyramid lens) ofthe lens array of the lens array sheet according to the first to thirdaspects of the present invention;

FIG. 29 is a perspective view for explaining a two-layer structure ofthe lens array sheet according to the first to third aspects of thepresent invention;

FIG. 30 is a perspective view showing an edge light type surface lightsource according to an embodiment of the first to third aspects of thepresent invention;

FIG. 31 is a perspective view showing an edge light type surface lightsource according to an embodiment of the first to third aspects of thepresent invention;

FIG. 32 is a perspective view showing an edge light type surface lightsource according to another embodiment of the first to third aspects ofthe present invention;

FIG. 33 is a conceptual schematic diagram showing an example of afabrication apparatus of the lens array sheet according to the first tothird aspects of the present invention;

FIG. 34 is a schematic diagram for explaining a microscopic action oflight that travels from the inside of an optical conductor to theoutside;

FIG. 35 is a schematic diagram for explaining a microscopic action oflight that travels from the optical conductor to the lens array sheetspaced apart therefrom for a small distance corresponding to tunneleffect; and

FIG. 36 is a schematic diagram for explaining that the contact portionof the cluster of the lens array sheet according to the first to thirdaspects of the present invention and the front surface of the opticalconductor contributes the distribution of the light in the opticalconductor.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

First, with reference to FIGS. 1 to 6, a lens array sheet according to afirst aspect of the present invention will be described.

FIG. 1 is a perspective view showing a lens array sheet according to thefirst aspect of the present invention. A lens array sheet 1 according tothe first aspect of the present invention shown in FIG. 1 is composed ofa transparent substrate sheet 31, a lens array 4, and a cluster 2. Thelens array 4 is composed of a large number of triangular prisms as lenselements 41 that are adjacently and one-dimensionally arranged on afirst surface of the transparent substrate sheet 31 so that the edgelines of the lens elements 41 are arranged in parallel. The cluster 2 iscomposed of a large number of cluster members 21 that are formed in arectangular parallelepiped shape and randomly and two-dimensionallyarranged on the entire second surface (rear surface) of the transparentsubstrate sheet 31. In FIG. 1, the cluster 2 is drawn on the secondsurface (front surface) for convenience of drawings.

The lens array sheet according to the first aspect of the presentinvention features the cluster formed on the opposite surface of thelens array. The cluster is composed of a large number of cluster membersthat are formed in a prism shape. The length of each side of eachcluster member is in the range from the wave length of the light of thelight source to 500 μm. The cluster members are randomly andtwo-dimensionally disposed on the entire surface of the second surfaceof the lens array sheet.

Each cluster member 21 is formed in a prism shape. The prism shapeincludes a triangular prism, a quadrangular prism, a pentagonal prism,and a hexagonal prism. The quadrangular prism includes a rhombus-shapeprism, a square-shape prism, and so forth. Among these shapes, aquadrangular prism formed in a rectangular parallelepiped shape ispreferable from a view point of easy fabrication.

FIG. 2 is a perspective view showing a cluster member 21 formed in arectangular parallelepiped shape. The relation among the height H, thewidth a, and the depth b may be either a=b=H (cube), a=b≠H, a≠b=H, ora≠b≠H.

As exemplified in FIG. 3, when the lens array sheet is disposed, thecluster members function as spacers for another lens array sheet or thelight emitting surface of the optical conductor. In the case that theheight H of each cluster member on the entire surface of the lens arraysheet is the same, when the lens array sheet is disposed, it is not bentand homogeneous distance can be maintained. Thus, Newton rings hardlytake place. The width a and the depth b can be varied for each clustermember. However, it is preferable to designate the same value to each ofthe width a and the depth b for each cluster member.

The size of each of the height H, the width a, and the depth b ispreferably in the range from the wave length of the source light to 500μm, more preferably to 125 μm. When the source light has a spectrumdistribution, the size of each of the height H, the width a, and thedepth b should be homogeneous to or greater than the maximum wave lengthof the spectrum of the visible light. When the size of each of theheight H, the width a, and the depth b is smaller than the wave lengthof the light, the occurrence of the equal-thickness interference fringesor the unification due to the optical contact of the lens array sheetand the optical conductor cannot be effectively prevented. On the otherhand, when the size of each of the height H, the width a, and the depthb exceeds 500 μm, the lens array sheet tends to bend and deform. Thus,moire fringes tend to take place between the pixels of the displaydevice or the fabrication of the lens array sheet becomes difficult.Consequently, it is worthless to increase the size.

Next, the reason that the height H of each cluster member 21 should behomogeneous to or greater than the wave length of the source light willbe described.

Assume the case that the lens array sheet is disposed on a plate-shapeoptical conductor so that the rear surface of the lens array sheetcontacts the front surface of the optical conductor.

As shown in FIG. 34, when incident light L₁ that travels from the insideof an optical conductor 51 to air reaches the front surface 55 of theoptical conductor that is an interface between the optical conductor 51and air, if the incident angle θ is greater than the critical angle θc,a total reflection takes place. Thus, all the energy of the incidentlight L₁ becomes reflection light L_(1R), not travels into the air.

However, when this phenomenon is microscopically observed, theelectromagnetic field of the incident light permeates from the frontsurface 55 of the optical conductor to the air for a distance of aroundthe wave length λ of the source light due to the tunnel effect. Theintensity of the permeated tunnel effect electromagnetic field L_(1V)attenuates exponentially in proportion to the permeated distance. Whenthe light travels in the air for the distance of around the wave lengthλ, it returns to the optical conductor 51 side. Thus, when thisphenomena is macroscopically observed, all the light energy is reflectedon the front surface 55 of the optical conductor.

Thus, as shown in FIG. 35, when the distance ΔX between the lens arraysheet 1 and the front surface of the optical conductor approaches thedistance smaller than the wave length of the source light (ΔX<λ), thetunnel effect electromagnetic field L_(1v) that has not been completelyattenuated becomes a progressive wave (output light) in the lens arraysheet 1. Thus, the light travels in the lens array sheet 1.

Thus, when the height H of each cluster member 21 is smaller than λ, thelight emitted from the optical conductor is not totally reflected on theentire front surface 55 of the optical conductor. Thus, as with incidentlight L₁, L₂, and L₃ shown in FIG. 36, rays of light in the vicinity ofthe light source that have most of light energy of the light source areemitted from the front surface of the optical conductor regardless ofwhether or not their incident angle θ is smaller than the criticalangle. Thus, the luminance of the output light in the vicinity of thelight source is high due to the presence of the light L₁ to L₃. However,the intensity of rays of light that are distributed to portions awayfrom the light source is weak. As with light L₄, since the distancebetween the light and the front surface of the optical conductor is longand the intensity thereof is attenuated, the luminance of the light inthe vicinity of the light source is high. In other words, the luminanceof the light apart from the light source is low.

On the other hand, when the height H of each cluster member 21 ishomogeneous to or greater than λ, namely

    H≧λ                                           Formula 1!

even in the vicinity region of the light source, at a portion of whichthe cluster member 21 contacts the optical conductor 51, as with thelight L₃ of FIG. 36, rays of light with incident angles smaller than thecritical angle travel from the front surface 55 of the opticalconductor. On the other hand, part of incident light L₁ and L₂ of withincident angles θ homogeneous to or greater than the critical angle aretotally reflected and transmitted to a light reflection layer on the farside of the optical conductor. Part of light that is diffusion-reflectedare entered into the front surface 55 of the optical conductor with anincident angle smaller than the critical angle. In other words, as withthe light L₁ of FIG. 36, light entered into the portion of which thecluster member 21 contacts the front surface 55 of the optical conductorbecomes output light as with L_(1T). On the other hand, the light thatenters a space portion 9 with a length homogeneous to or greater thanthe wave length λ of the light is totally reflected on the front surface55 of the optical conductor and distributed to the far region. Thus, ina region far from the light source, the amount of the light emitted fromthe front surface of the optical conductor is satisfactorily high.

As described above, when the lens array sheet with the cluster members21 whose height H is homogeneous to or greater than λ is disposed on theoptical conductor so that the lens array sheet faces the front surfaceof the optical conductor, light with a homogeneous luminancedistribution in the entire region on the front surface of the opticalconductor can be obtained.

Next, the structure of which two lens array sheets are layered will bedescribed. As was proposed with Japanese Patent Laid-Open PublicationNo. 5-323214 by the applicant of the present invention (not disclosedwhen the application of the present invention was filed), the conditionfor causing the equal-thickness interference fringes to disappear isexpressed by the following formula.

    H≧λ/(2Δφ.sup.2)                     Formula 2!

where H is the height of each cluster member 21; λ is the wave length ofan outer source light; and φ is the angular range of visual field of theouter light source (such as sun light through a window, light of a lampon the ceiling) reflected on the light reflection surface (frontsurface, rear surface, or the like) of the lens array sheet.

However, in the condition of which the conventional lens array lens isused, when the Formula 1! is satisfied, the Formula 2! is alsosatisfied. In other words, λ/(2Δφ²)≧λ is satisfied.

Next, practical values of the Formula 2! are obtained. The front surfaceof the lens array sheet is observed with white light with a wave lengthin the range of 0.38 μm≦λ≦0.78 μm as external light. In addition, whenthe angular radius of the external light source as indoor light ornatural light through a window is in the range of 10°≦Δφ≦120°, namely0.175 rad!≦Δφ≦2.094 rad!. Thus, corresponding to Δφ=0.175 rad! andλ_(MAX) =0.78 μm!, the following relation can be obtained.

    H≧12.5  μm!>λ.sub.MAX =0.78  μm!

Each of the width a and the depth b of each cluster member 21 ispreferably the same as the height H. However, to maintain the minimumstrength of the cluster member 21 as a spacer, each of the width a andthe depth b should be 1 μm or greater although they depend on the heightH. When each of the width a and the depth b exceeds 125 μm, inparticular, 500 μm, the cluster member becomes visible. When the lensarray sheet is used for a liquid crystal display device, the moirefringes take place in pixels of the liquid crystal display device.

The cluster members 21 with the above-described size on the lens arraysheet are preferably randomly and two-dimensionally distributed. If thecluster members are periodically formed, the cluster members and lenselements (that are periodically formed) formed on the opposite surfacethereof are periodically overlaid. Thus, the moire fringes take place.In addition to the periodical arrangement of the lens elements of thelens array, when the lens array sheet is used for a back light of acolor liquid crystal display device, the cluster members interfere withthe periodical arrangement of pixels of the display device and therebymoire fringes tend to take place. Thus, when the cluster members arerandomly arranged, the moire fringes can be prevented from taking place.

However, even if the cluster members 21 are randomly arranged, when theshape of each cluster member 21 is the same and the orientation thereofis the same, since each side surface of the same type of the clustermembers faces the same direction, a set of small side surfaces in thesame direction form a large virtual side surface. Since the virtual sidesurface is composed of cluster members that are randomly arranged, theydo not have a periodicity. However, since the virtual side surfaceinterferes with the lens elements of the lens array, the moire fringestake place.

Thus, the structural surface of each lens element and the side surfaceof each cluster member should have a particular relation.

FIGS. 4A and 4B are schematic diagrams for explaining a structure forpreventing the moire fringes from taking place. For example, as shown inFIG. 4A, assume a structure of which the lens array of the lens arraysheet 1 is composed of triangular prism lenses as lens elements 41. Thelight emitting surface of the lens array sheet 1 is in parallel with theX-Y plane. The light emitting surface is referred to as a horizontalsurface. The normal direction perpendicular to the light emittingsurface is the direction of the axis Z (not shown). The structuralsurfaces of each lens element 41 are inclined surfaces 42 that form thetop and the bottom of a triangular prism. The line of intersection ofthe inclined surfaces and the horizontal surface is in parallel with theaxis X (in this case, the coordinates are defined so that the axis X isin parallel with the line of intersection). Strictly speaking, theinclined surface is a finite surface. The horizontal surface can bedefined in various manners depending on the coordinates of the axis Z.The inclined surface does not intersect with the horizontal surfacedepending on a condition. In this example, the line of intersectionrepresents the line of which the inclined surface is extended andintersected with the horizontal surface. When triangular prisms as lenselements are one-dimensionally arrayed, there is one line ofintersection. On the other hand, when quadrangular prisms as lenselements are two-dimensionally arrayed, there may be two or more linesof intersection. In this case, the lines of intersection may be notperpendicular to each other.

FIG. 4B is a schematic diagram showing the case that X-Y coordinatescorresponding to the line of intersection of the lens elements 41 of thetriangular prisms is overlaid with X'-Y' coordinates corresponding toaxis X' of one line of intersection obtained from the cluster 2.

The orientations of the cluster members 21 (formed in a rectangularparallelepiped shape) are arranged. There are two lines of intersectionof side surfaces of the cluster members 21 and the horizontal surface ofthe lens array sheet. The two lines of intersection are perpendicular toeach other. They are lines of intersection in parallel with the axis X'and the axis Y'. The axis X' and the axis X form an angle α.

There are many dispersed cluster members. In addition, there are manylines of intersection of the many side surfaces and the horizontalsurface of the lens array sheet. However, since the orientations of thecluster members are arranged, in the case of the rectangularparallelepiped shape, there are two lines of intersection that areperpendicular to each other.

When the angle α between the axis X and the axis X' is zero, the axis Xis in parallel with the axis X'. Thus, the moire fringes tend to takeplace. However, when the line of intersection of each lens element hasan angle of 5° to the line of intersection of each cluster member, themoire fringes can be prevented. In other words, in the case of therectangular parallelepiped shape, when the angle α is in the range from5° to 85° (in the clockwise direction), more preferably in the rangefrom 10° to 80° (in the clockwise direction), the moire fringes can beeffectively prevented. In addition, the angle α is preferably in therange from -5° to -85° (in the counterclockwise direction), morepreferably in the range from -10° to -80° (in the counterclockwisedirection). In the case of the rectangular parallelepiped shape, whenthe angle α exceeds 85°, the angle to the line of intersection of theside surface becomes large. Thus, since the relation between theadjacent side surfaces (90° to the side surface) becomes almostparallel. Consequently, due to the relation with the adjacent sidesurfaces, the moire fringes tend to take place. When the side surfacesof the prisms have an angle of 5° to the horizontal direction, the moirefringes can be prevented.

When the cluster members are composed of for example rectangularparallelepiped members and the angle between the line of intersection ofa particular side surface of each rectangular parallelepiped member andthe horizontal surface of the lens array sheet and the line ofintersection of the surface of each lens element and the horizontal lineexceeds 5°, it is not necessary to arrange the orientations of all thecluster members (formed in the rectangular parallelepiped shape). Forexample, even if 1% of all cluster members are arranged in parallel,when they are not adjacently arranged as a set, the parallel relation ofwhich the moire fringes take place is not defined.

Thus, in claim 3 of the first aspect of the present invention, "eachrectangular parallelepiped members" where the line of intersection of aside surface of each rectangular parallelepiped member is not inparallel with the line of intersection of a lens element does not meanthat all rectangular parallelepiped members that are formed do not havea non-parallel relation, but that even if part of rectangularparallelepiped members have a parallel relation, the non-parallelrelation takes place as the general situation.

As the shape of each cluster member according to the first aspect of thepresent invention, a prism shape can be used instead of the rectangularparallelepiped shape. In the above-described rectangular parallelepipedshape, the angle of each adjacent side is 90°. Thus, whenever therectangular parallelepiped members are rotated for 90°, the samesituation takes place. However, in the case of the rectangularparallelepiped shape, since each opposite side surfaces are in parallel,it is necessary to consider two lines of intersection that areperpendicular to each other. However, in the case of the prism shapeother than the rectangular parallelepiped shape, for example atriangular prism shape, the number of lines of intersection to beconsidered is three. In the case of a pentagonal prism shape, the numberof lines of intersection to be considered is five. In these cases, thenumber of lines of intersection to be considered is greater than that inthe case of the rectangular parallelepiped shape. Thus, the probabilityof the occurrence of the moire fringes increases. Consequently, thedegree of freedom of designing the lens array sheet decreases. Even inthe case of a free quadrilateral shape where each adjacent side does notform a right angle, the number of lines of intersection to be consideredis as many as four. Thus, even if a quadrangular prism shape with a rearsurface that has a parallelogram shape or a rhombus shape is used, aswith the case of the rectangular parallelepiped shape, the occurrence ofthe moire fringes can be prevented. However, the cluster members in therectangular parallelepiped shape is more easily fabricated than those inthe quadrangular prism shape with the rear surface having aparallelogram shape or a rhombus shape.

In the case that the lines of intersection of side surfaces are notstraight lines, there is an n-side prism (where n is infinite) (namely,a circular cylinder shape or an elliptic cylinder shape where the sidesurface is a curved surface). In this case, when an original press filmfor forming cluster members is produced by a horizontal scanning methodusing a scanner or the like, since the cluster members are very small,the contour of for example a circular shape of side surfaces that arenot in parallel with or perpendicular to scanning lines are rugged.Thus, a smooth side surface of the cylinder cannot be obtained.

As a method for randomly forming cluster members, X-Y coordinates onwhich cluster members are arranged corresponding to random numbers in anX-Y plane with a predetermined area equivalent to the entire surface ofthe lens array sheet are generated. In FIG. 5A, reference numeral 22 isa random coordinate point at which a cluster member 21 is arranged.

When cluster members 22 with a finite size are adjacently formed in thecoordinate points 22, overlap portions 23 of the cluster members 22 maytake place as shown in FIG. 6A. In FIG. 6A, dashed lines are virtuallines that represent overlap portions. In this case, since the size ofcluster members becomes large and thereby visible. As one method forsolving such a problem, as shown in FIG. 6B, the height of the overlapportions of cluster members is preferably zero. Thus, the clustermembers can be prevented from overlapping and increasing the area of thetop portion thereof. Consequently, even if the cluster members overlap,the size thereof can be prevented from increasing and the clustermembers can be prevented from becoming visible. FIG. 5B shows theoverlap portions. FIG. 5C shows the cluster members in the case that theheight H of the overlap portion is zero.

The moire fringes that take place in the relation between the structuralsurface of each cluster member and the structural surface of each lenselement. In other words, when the cluster members are formed in the sameorientation, the side surfaces thereof are arranged. Thus, a line ofintersection that can be recognized is defined. This is because therelation between the line of intersection of each cluster member and theline of intersection of each lens element takes place. However, even ifthe shapes of the cluster members are the same, when they are randomlyformed (namely, unlike with the case shown in FIG. 4B, the clustermembers are rotated around the axis Z that is perpendicular to the X-Yplane), the line of intersection of a side surface of each clustermember has an angle that is dispersed. Thus, there is no line ofintersection defined at a predetermined angle. In such a manner, theoccurrence of the moire fringes can be prevented. However, from a viewpoint of easy fabrication of the lens array sheet, it is preferable toform the cluster members in the same orientation.

In this point, the circular cylinder shape, the elliptic cylinder shape,and the like are superior to the other shapes. However, as describedabove, the side surface that is a smoothly curved surface is difficultto fabricate. As a countermeasure in the case that adjacent clustermembers overlap, when the height H is zero, a sharp sectional shape isformed at a contact portion. This shape causes the fabrication of thelens array sheet to become difficult.

However, instead of using the method of which the height H is zero, whenX and Y coordinate values are generated corresponding to random numberswith a quantizing step that is greater than the diameter D of forexample a circular cylinder (values smaller than the quantizing step arerounded off), the random coordinate values are always greater than thediameter D. Thus, even if the cluster members are formed at thecoordinate points, they do not overlap. As a modification of thismethod, when the quantizing step is intentionally increased, the minimumdistance of each adjacent cluster member can be adjusted.

The distribution density of the cluster members is designated so thatthe lens array sheet is not bent and thereby the equal-thicknessinterference fringes do not take place. In addition, even if the lensarray sheet has a proper rigidity, a homogeneous distance between thelens array sheet and the optical conductor or between the lens arraysheets can be maintained so that a small difference of the distancesprevents the equal-thickness interference fringes from taking place.

In the case that two lens array sheets are layered, the distributiondensity of which the sectional area of each cluster member is zero(namely, the distribution density of the cluster members) is preferablydesignated to the relation of t<2p (where t is the average distance ofadjacent cluster members formed on the rear surface of the upper lensarray sheet; and p is the repetitive period of the lens elements formedon the front surface of the lower lens array sheet). Thus, sincesupporting contacts between the cluster members 21 formed on the rearsurface of the upper lens array sheet and the lens elements 41 formed onthe front surface of the lower lens array sheet are prevented from beingbent, the distance between the upper and lower lens array sheets doesnot become heterogeneous. Consequently, the equal-thickness interferencefringes do not take place. In addition, the distance between the upperand lower lens array sheets can be prevented from becoming smaller thanthe wave length of the source light. The average distance t is morepreferably in the range of t<0.5 p.

On the other hand, as a distribution density for preventing theequal-thickness interference fringes from taking place even if the lensarray sheet bends in the case that the sectional area of each clustermember is finite, the area ratio Sr (=(Sp/St)×100) of the sum of Sp ofthe sectional areas of the cluster members against the entire area St ofwhich the lens array sheet 1 faces the optical conductor 51 ispreferably in the range from around 0.01 to 60%. As the function ofspacers, the number of cluster members should be as small as possible.However, to prevent the lens array sheet from bending, a proper numberof cluster members are required. When the lens array sheet is used as asurface light source along with an optical conductor (that will bedescribed later), a proper number of cluster members are required tohomogenize the surface distribution of the luminance.

Next, the factor of the surface distribution of the luminance will bedescribed using the area ratio R that is the reverse relation of theabove-described area ratio Sr.

The area ratio R %! that is the ratio of the sum Sa of the areas ofspace portions 9 with a length homogeneous to or greater than the wavelength of the source light (the cluster members 21 do not contact thefront surface of the optical conductor 51) and the entire area St (thelens array sheet 1 faces the optical conductor 51) is expressed by thefollowing Formula 3!.

    R=(Sa/St)×100                                         Formula 3!

Thus, the area ratio R and the area ratio Sr have the relation ofR+Sr=100.

The area ratio R depends on the homogeneity of the luminance on thesurface to which light is emitted, the using efficiency of the lightenergy, the size of the optical conductor, and so forth. The area ratioR should be normally 80% or greater, preferably 90% or greater.

As the reason, when the smooth front surface 55 of the optical conductoris contacted with the front surface (the rear surface) of the lens arraysheet 1 in the case that the surface roughness of both the opticalconductor 55 and the lens array sheet 1 are homogeneous to or smallerthan the wave length of the source light, as shown in FIG. 30, most oflight entered from the light source 52 to the optical conductor 51 arenot totally reflected in a region from the side edge portion of thelight source side to the distance y, but emitted (even if the light isentered into the front surface of the optical conductor with an incidentangle homogeneous to or greater than the critical angle, the light isnot totally reflected, but emitted to the lens element). Thus, at aposition spaced apart from the light source by greater than the distancey, the luminance remarkably decreases and becomes dirk.

Experimental results show that the percentage of the distance y of thelight emitting portion to the entire length Y in the light propagatingdirection is in the range from 10 to 20%.

Thus, to homogeneously distribute the light energy entered from thelight source to the optical conductor in the entire length Y, since mostof light (approximately 100%) is emitted at the region of the length yof the front surface 55 of the optical conductor, 10 to 20% of theincident light of the region of the length y should be emitted and therest of the incident light (namely, 90 to 80% of the light) should betotally reflected.

Since the following relation is satisfied (amount of totally reflectedlight/amount of entire incident light)≈Sa/St=R thus, R should be 80 to90% (Sr=10 to 20%).

Since the similar approximation can be performed at a position fartherthan y, the condition of which R should be in the range of 80 to 90% canbe applied to the entire length. However, when R becomes almost 100%(namely, Sr becomes almost 0%), as described above, since the lens arraysheet is bent, the distance of each cluster member cannot be maintainedto homogeneous to or greater than the wave length of the source light.Thus, the upper limit of R is preferably 99.99% or smaller (namely,Sr≧0.01%).

When the above-described cluster members are formed on one surface ofthe lens array sheet, light that is emitted out of the angular range ofvisual field is not increased and thereby the luminance is notdecreased. In addition, the equal-thickness interference fringes and themoire fringes can be prevented. Thus, the lens array sheet canhomogeneously distribute light on the entire surface of the opticalconductor with a homogeneous surface distribution.

Next, with reference to FIG. 7, a lens array sheet according to a secondaspect of the present invention will be described. The second aspect ofthe present invention features the random two-dimensional distributionof the first aspect of the present invention is a distribution of whichthe positions of lattice points of a two-dimensional periodic latticeare randomly moved and reallocated. For simplicity, redundantdescription is omitted. Only portions peculiar to the second aspect ofthe present invention will be described.

As a method for randomly forming cluster members, X and Y coordinatesfor the cluster members may be generated on the X-Y plane with apredetermined area equivalent to the entire surface of the lens arraysheet corresponding to random numbers. In this case, distribution errorsthat are visible may take place in the distribution of coordinatepoints.

To solve such a problem, in the lens array sheet according to the secondaspect of the present invention, coordinate points at which the clustermembers are formed are generated corresponding to a predetermined rule.Regardless of the number of cluster members, they are randomlydistributed with a homogeneous surface density of cluster memberswithout a deviation.

In other words, according to the second aspect of the present invention,coordinate points at which cluster members are formed are not randomlygenerated. Instead, coordinate points are regularly and homogeneouslygenerated (periodically generated), and then the coordinate points arerandomly moved and reallocated. Thus, the coordinate points are randomlygenerated. Consequently, the number of cluster members in apredetermined area (namely, the surface density of cluster members)depends on the number of cluster members that have been regularlygenerated.

The coordinate points regularly and homogeneously generated are obtainedcorresponding to periodic lattice points 8 of two-dimensional periodiclattices as shown in FIGS. 7A to 7C. A two-dimensional lattice iscomposed of lattice elements that are adjacently, periodically, andtwo-dimensionally arranged. Next, lattice elements of thetwo-dimensional lattices will be described. FIG. 7A shows a squarelattice of which the lattice constant a on the axis X and the latticeconstant b on the axis Y are the same on an orthogonal coordinatesystem. As another two-dimensional lattice, the lattice length a and thelattice length b of each lattice element are not the same. As anothertwo-dimensional lattice, the coordinate axes are not perpendicular toeach other. These two-dimensional lattices are referred to asparallelogram lattices. FIG. 7B shows a basket-weave-shape lattice. FIG.7C shows a hexagonal lattice.

Next, a method for randomly moving and reallocating the lattice pointsof an above-described two-dimensional lattice will be exemplified. Forexample, in the two-dimensional lattice shown in FIG. 7A, the Xcoordinate Px and the Y coordinate Py of the coordinates P (Px, Py) ofeach lattice point can be expressed by the following Formula 4.1!. Thecoordinates R (Rx, Ry) of a new lattice point that has been randomizedcan be expressed by the following Formula 4.2! and Formula 4.3!.

    Px=a×n, Py=b×m                                  Formula 4.1!

    Rx=a×n+α×RND                              Formula 4.2!

    Ry=b×m+β×RND                               Formula 4.3!

where a is the length of the lattice element on the axis X; b is thelength of the lattice element on the axis Y; n and m are any positive ornegative integer including zero; RND is a random function for generatinga uniform random number in the range from -0.5 to +0.5; and α and β arecoefficients multiplied by the obtained random number. Alternatively,the random number may be in the range from 0 to 1. Thereafter, when 0.5is subtracted from the resultant random number, the same result isobtained.

When α and β are smaller than a and b, respectively, lattice points thathave been randomized are reallocated in the vicinity of the originallattice points. When α and β approach zero, the randomized latticepoints are asymptotically converged to the original lattice points andthereby the random characteristic is asymptotically converged. On theother hand, when α and β are greater than a and b, respectively, theregularity of the original lattice points degrades. Thus, when α and βare proper values to a and b, respectively, random coordinate pointswith both regularity and randomness can be obtained. FIG. 7D showsrandomized lattice points in the case that α=1.4a and β is 1.4b. FIG. 7Eshows randomized lattice points in the case that α=0.4a and β=0.4b. InFIGS. 7D and 7E, cluster members 2 formed at randomized coordinatepoints in a square are shown.

When cluster elements are formed at randomized lattice points, forexample the center of each cluster member with a finite area as itsrepresentative coordinates can be positioned at a lattice point. Thus,all cluster members should be formed at representative coordinates ofindividual lattice points corresponding to the same definition.Otherwise, a new element is added to the coordinates of the randomizedlattice points. Thus, unless a special condition is added, therandomness cannot be maintained. The size of the rear surface (namely,the bottom area) of each cluster member should be designated so that itsatisfies an area ratio R (that will be described later).

The above-described randomizing process can be performed by for examplea computer. The calculated result is printed on an original press film.Alternatively, a block of cluster members that have been randomized canbe repeatedly formed in vertical and horizontal directions so as to formcluster members with a required area. This process can be accomplishedby a known printing method.

The distribution density of the cluster members should be designated sothat the lens array sheet does not bend and thereby the equal-thicknessinterference fringes do not take place. In addition, even if the lensarray sheet has a rigidity to some extent, the distribution density ofthe cluster members should be designated so that the distance betweenthe optical conductor and the lens array sheet or between two lens arraysheet can be homogeneously kept and thereby the equal-thicknessinterference fringes do not take place. Thus, the size of each latticeelement of the two-dimensional lattice can be designated correspondingto the distribution density.

As described above, according to the second aspect of the presetinvention, since the predetermined cluster members are formed on onesurface of the lens array sheet, the equal-thickness interferencefringes and moire fringes can be prevented without tradeoffs of increaseof the amount of light that is emitted out of the angular range ofvisual field and decrease of the luminance. In addition, a lens arraysheet that can homogeneously distribute light on the entire frontsurface of the optical conductor.

Next, with reference to FIGS. 8 to 20, a lens array sheet according to athird aspect of the present invention will be described. For simplicity,redundant description is omitted.

FIG. 8 is a perspective view showing a lens array sheet according to anembodiment of the third aspect of the present invention. A lens arraysheet 1 according to the third aspect of the present invention shown inFIG. 8 is composed of a transparent substrate sheet 31, a lens array 4,and a cluster 2. The lens array 4 is composed of a large number oftriangular prisms as lens elements 41 that are adjacently andone-dimensionally formed on a first surface of the transparent substratesheet 31 so that the edge lines of the lens elements 41 are arranged inparallel. The cluster 2 is composed of a large number of cluster members21 that are formed in a rectangular parallelepiped shape and randomlyand two-dimensionally formed on the entire second surface of thetransparent substrate sheet 31. In FIG. 8, the cluster 2 is formed onthe second surface (front surface) for convenience.

The lens array sheet according to the third aspect of the presentinvention features a cluster formed on the opposite surface of the lensarray. The cluster is composed of a large number of cluster members thatare separately formed with different shapes. Each of cluster members iscomposed of at least one cluster member element that is formed in aprism shape or a prismoid shape and whose minimum diagonal length of thefront surface and rear surface is homogeneous to or greater than thewave length of the source light and whose maximum diagonal lengththereof is 500 μm or smaller. At least one cluster member element isfused. The fusing of cluster member elements is performed by connectingadjacent cluster member elements. To obtain cluster members from clustermember elements, in the third aspect of the present invention, a specialmethod is used. In other words, in the third aspect of the presentinvention, cluster members are formed of cluster member elementscorresponding to the theory or percolation. In this theory, clustermember elements are disposed at lattice points of a two-dimensionallattice such as a square lattice with a predetermined probability (thatis referred to as an occupying probability P or percolation probabilityP). Cluster member elements disposed at adjacent lattice points areconnected and thereby a plurality of cluster members are formed. Whenthere are no adjacent cluster member elements, each cluster memberelement is treated as a cluster member. In addition, as will bedescribed later, cluster members are disposed out of lattice points.

Next, the method for obtaining cluster members corresponding to thetheory of percolation will be described.

First, the theory of percolation will be described. There are two typesof randomness (see "Percolation" in Fractal, Chapter 9, written by J.Feder, translated by Mitsugu Matsushita, et. al, published by KeigakuSyuppan on May 31, 1991). The first type is known as a diffusionphenomenon that is randomness of particles that move in a medium. Thesecond type is randomness of particles that move in a mediumcorresponding to the randomness thereof. The latter randomness is namedpercolation process by Hammersley because the particles move as withcoffee in a percolator. A real example of the latter randomness is thepenetration of water in a crack of a rock or a stone. In addition, aburning process of a fire is explained with such a model in "Propagationof fire and percolation model" in SUURIKAGAKU, by Motoo Hori, pp 63-70,June 1974.

J. Feder explains the concept of the percolation with a two-dimensionalpenetration on a square lattice. In this case, the lattice points of thesquare lattice are randomly occupied with particular substances (forexample, cluster member elements in the third aspect of the presentinvention) at a occupying probability P (this is called percolationprobability). In this case, assumed that "particular substances" aresmall holes in a rock or a stone. In addition, assume that adjacentholes are connected with a pipe. (In this case, a small hole is referredto as a site. A set of connected sites is referred to as a cluster. Apipe that connects sites is referred to as a bond.) Thus, water pouredto a particular hole penetrates to a small hole that is connected to theparticular hole. In other words, water poured to a small hole thatcomposes a particular cluster stays in the cluster, not furtherpenetrates. The number of sites that compose one cluster is referred toas the size of the cluster. However, when the occupying probability Pexceeds a predetermined critical value, an infinite cluster that is aset of small holes takes place. Water poured to a small hole of theinfinite cluster penetrates to the entire lattice. Such a cluster isreferred to as a penetrate cluster. The occupying probability of whichthe percolation cluster takes place at first is referred to as acritical percolation concentration Pc (or a critical probability). Inthe case of a square lattice, Pc≈0.593.

In the above-described example, when lattice points are occupied with apredetermined probability and the occupied lattice points are adjacent,they are connected with pipes. A model represented only with theoccupying probability of the lattice points is referred to as a sitepercolation. A model represented with the probability of which aconnected pipe is open (not closed) is referred to as a bondpercolation. In addition, a model represented with both the probabilityof site and the probability of bond is referred to as a mixedpercolation. In the technical paper written by Motoo Horii, thepropagation of a fire is evaluated with such percolation models. As anexample of the bond percolation, a phenomenon of which a fire of awooden house in a thickly housed area is extended to an adjacent housedepends on the bond probability. In addition, the value of theconcentration Pc of the critical percolation is used for the sitepercolation of the square lattice. The value of the concentration Pc ofthe bond percolation is 1/2.

Next, the relation between the cluster member elements and the clustermembers according to the third aspect of the present invention will bedescribed corresponding to the explanation of the clusters of theabove-described percolation models.

In the site percolation of the square lattice, when lattice points ofthe square lattice are viewed from the obtained cluster, lattice pointsthat compose the cluster have small holes. Lattice points that do notcompose the cluster do not have small holes. Thus, in the third aspectof the present invention, lattice points that compose a cluster (namely,lattice points in a cluster) are referred to as cluster structuralelements. Consequently, lattice points that do not have small points(namely, lattice points outside the cluster) are not referred to ascluster structural elements. In other words, lattice points arealternatives of which substances can be designated with a predeterminedprobability. Cluster structural elements are alternatives to whichsubstances are always designated.

In the site percolation, occupied lattice points are sites. A cluster isobtained by connecting adjacent "sites occupied with particularsubstances" with bonds. This definition can apply to the bondpercolation and the mixed percolation. In the bond percolation, acluster is obtained by connecting "bonds occupied with particularsubstances" with sites. In the mixed percolation, assuming that allbonds are not connected, but part of bonds are connected with apredetermined probability, although a percolation cluster hardly takesplace, a cluster can take place.

Thus, in the bond percolation, a cluster is composed of particular (forexample, occupied) bonds. In the mixed percolation, a cluster iscomposed of particular sites or bonds. In addition, a cluster is notlimited to the site percolation model. According to the third aspect ofthe present invention, cluster structural elements are sites, bonds, orthe both regardless of the site percolation model, the bond percolationmodel, or the mixed percolation model.

However, in the following description, a cluster in the site percolationmodel will be exemplified. In the third aspect of the present invention,cluster members are obtained by connecting a plurality of cluster memberelements. Thus, the site percolation model, where cluster memberelements are disposed at lattice points with a predeterminedprobability, is intuitively understandable. In addition, logically, ithas been proved that the bond percolation model is equivalent to thesite percolation model (where the center of each bond is placed at alattice point (this lattice is referred to as a coated lattice)) (see"Science of Percolation (translated title)", by Takashi Odagaki, Chapter1, Syokabo, Jun. 20, 1993).

A typical example of a two-dimensional lattice is a square lattice.However, the two-dimensional lattice is not limited to the squarelattice. Normally, a two-dimensional lattice is a lattice with aregularity where lattice elements are periodically and two-dimensionallyformed. Examples of the two-dimensional lattices are a square lattice, atriangular lattice, a basket-weave-shape lattice (see FIG. 14A), ahexagonal lattice (hive) (see FIG. 14B), and a Penrose lattice. A squarelattice is composed of square lattice elements where the lattice lengthon the axis X is homogeneous to the lattice length on the axis Y in theorthogonal coordinate system. Instead of the square lattice, aparallelogram lattice or the like where the lattice length on the axis Xis not homogeneous to the lattice length on the axis Y or where thecoordinate axes are not perpendicular to each other (namely, an obliquecoordinate system) may be used. Alternatively, lattices with irregularlattice elements may be used.

The critical percolation concentration Pc in the site percolation modelof each of major two-dimensional lattices is as follows.

Hive lattice: 0.6962

Square lattice: 0.592745

Basket-weave-shape lattice: 0.65271

Triangular lattice: 0.5

Penrose lattice: 0.584

Although the shape of the cluster is random, it is normally fractal.

A cluster member element that is occupied at a lattice point with apredetermined probability is formed in a prism shape or a prismoidshape. The rear surface of the cluster member element is formed in apolygon shape that is for example a triangle shape, a quadrilateralshape, or a hexagon shape. Examples of the quadrilateral shape are arectangle shape, a square shape, and a rhombus shape. The clustercomposed of quadrangular prisms formed in a rectangular parallelepipedshape where two opposite sides composing side surfaces thereof are inparallel with each other and adjacent side surfaces are perpendicular toeach other and formed in a rectangle shape or a square shape can beeasily fabricated.

The shape of the side surfaces of the cluster member elements that arealso the side surfaces of the cluster members is not limited to a prismshape. Instead, the shape may be a prismoid shape due to easyfabrication.

The shape of the rear surface of the prism member or prismoid membernormally should accord with the shape of the two-dimensional lattice ofa percolation process (that will be described later) due to symmetry. Inother words, when the two-dimensional lattice is a square lattice, thebottom shape is rectangular. When the two-dimensional lattice is atriangular lattice, the bottom shape is triangular. When thetwo-dimensional lattice is a hexagonal lattice, the bottom shape ishexagonal. However, the bottom shape is not limited to such shapes.

The minimum element of the cluster is an independent structural element.One minimum cluster member is formed by allocating one cluster memberelement to one structural element.

FIG. 9 is a schematic diagram showing a rectangular parallelepiped shapeas an example of the shape of the cluster member element 24. As thedefinition of the rectangular parallelepiped shape, the relation of theheight H, the width a, and the depth b may be either a=b=H (cubic),a=b≠H, a≠b=H, a=H≠b, or a≠b≠H≠a. In a rectangular parallelepiped shape,diagonal lines d1 and d2 on the front surface are the same as those onthe rear surface (not shown). The size of each cluster member elementsis preferably in a predetermined range corresponding to the height H andthe diagonal line d. The size of each of cluster member elementsallocated to a plurality of structural elements (in this example,lattice points) are normally the same.

The height H of a cluster member element is also the height of a clustermember composed thereof. As shown in FIG. 10, when a lens array sheet isdisposed, the cluster member elements function as spacers with anotherlens array sheet or a light emitting surface of an optical conductor.The height H of all the cluster members distributed on the entiresurface of the lens array sheet is preferably the same so that the lensarray sheet does not bend and thereby Newton rings do not take place.Thus, the height H of the cluster member elements disposed at latticepoints are also preferably the same. Thus, the height H of the clustermember elements disposed at the lattice points should be the same.

The size of the cluster member elements has an optimum range dependingon the wave length of the source light and the visible size ofsubstances. As factors of the size of the cluster member elements, theheight H, the width a, and the depth b are used. However, in this case,the cluster member elements are evaluated with the height H and thelength of the diagonal line d. As the diagonal line, the minimumdiagonal line and the maximum diagonal line of the front surface and therear surface (in the case of the prismoid shape, the diagonal line ofthe front surface is different from the diagonal line of the rearsurface) are evaluated.

Each of the height H and the length of the minimum diagonal line d ispreferably homogeneous to or greater than the wave length of the sourcelight. In addition, each of the height H and the length of the maximumdiagonal line d is preferably homogeneous to or smaller than 500 μm.more preferably homogeneous to or smaller than 125 μm. When the sourcelight has a spectrum distribution, each of the height H and the lengthof the minimum diagonal line d is homogeneous to or greater than themaximum wave length of the spectrum of the visible light.

When each of the height H and the length of the diagonal line d issmaller than the wave length of the light, the occurrence of theequal-thickness interference fringes or the unification due to theoptical contact of the lens array sheet and the optical conductor cannotbe effectively prevented. On the other hand, when each of the height Hand the length of the diagonal line d exceeds 500 μm, the lens arraysheet tends to bent and deform or the moire fringes tend to take placebetween cluster members and pixels of the display device. Alternatively,the fabrication of the lens array sheet becomes difficult. As a result,it is no meaning Lo increase the size of the cluster members.

Next, the length of the diagonal line of the cluster member elementswill be described. The minimum element of the cluster member 21 is acluster member element. Thus, as the size of the top and rear surfacesof the cluster member elements, the length of the diagonal line d shouldbe homogeneous to or greater than the wave length of the light source soas to maintain the minimum strength as spacers. However, the length ofthe diagonal line d is preferably 1 μm or greater. When the length ofthe diagonal line d exceeds 125 μm, in particular, 500 μm, the clustermembers or cluster becomes visible. Thus, the moire fringes tend to takeplace between cluster members and pixels of a liquid crystal displaydevice.

In addition, since cluster members are formed by fusing and connecting aplurality of cluster member elements, when the straight portion of theside surface of the fused cluster members is excessively large, theybecome visible although they depend on the regularity of thetwo-dimensional lattice, the number of fused cluster member elements,the complexity of the shape of the cluster member elements, the fusingmethod, and the complexity of the shape of the fused cluster members.Thus, the length of the straight portion is preferably homogeneous to orsmaller than 1000 μm. However, the shape of cluster members that areformed by fusing and connecting cluster member elements is at random.Thus, even if one cluster member has a very long straight portion on aside surface portion, it is not remarkable because other cluster membersdo not have such straight portions. When the two-dimensional latticetype, the occupying probability, the shape of cluster members, and thefusing method of the cluster members are properly selected, clustermembers in a desired shape can be obtained from the cluster memberelements.

The cluster member elements that are allocated to lattice points shouldnot have the same shape and the same size. Even if the shape and size ofthe cluster member elements are varied (for example, the cluster memberelements have the same shape, but three different sizes), as long as atwo-dimensional lattice with a regularity is used, the occurrences ofcluster members with a long straight portion can be reduced.

As a quantity for evaluating the spreading amount of the cluster, anaverage rotating radium Rs can be used as defined in the followingFormula 5!. ##EQU1## where r_(o) is the center-of-gravity positionvector of a cluster; r_(i) is the position vector of each site of thecluster; s is the size of the cluster; and Rs is the root of the averageof the square of the distance from the center of gravity of the clusterto each site of the cluster. Rs is preferably in the range of d≦Rs<3mm!. Although the lower limit is obvious, the upper limit is designatedcorresponding to the visible effect due to the optical contact.Otherwise, the equal-thickness interference fringes take place and theheterogeneity of the surface distribution of the output luminancebecomes remarkable.

Next, a real example for forming cluster members from cluster memberelements in a rectangular parallelepiped shape with a square rearsurface corresponding to the theory of percolation will be describedwith reference to FIGS. 12A and 12B.

FIG. 12A shows the case that cluster member elements 24 are allocated tolattice points (structural elements) of a 25×25 square lattice. Thelattice points are selected with an occupying probability P=0.4. In FIG.12A, black squares represent cluster member elements. The area of therear surface of each cluster member element is 5/8 of that of eachlattice element of the square lattice. Adjacent cluster member elements(connected with bonds) are fused by filling the gap therebetween. Inaddition, for adjacent four cluster member elements that form a square,a portion that surrounds the square is filled. Thus, as shown in FIGS.12A and 12B, cluster members 21 of which cluster member elements havebeen fused are obtained.

In this case, the occupying probability P is preferably smaller than thecritical percolation concentration Pc. When P≧Pc, a percolation clustertakes place as shown in FIG. 15. In FIG. 15, with P=0.60 that is thesame condition as that shown in FIG. 12, a percolation cluster 21 thatextend from the upper side to the lower side takes place. In the thirdaspect of the present invention, when the occupying probability Pexceeds the critical percolation concentration Pc, since the size of thecluster (that connects the end-to-end of the lattice although the sizeof the cluster is finite), the cluster tends to become visible. Inaddition, the area of the optical contact between the lens array sheetand the optical conductor or between the lens array sheets (the contactwith a space whose length is homogeneous to or smaller than the wavelength of the source light) cannot be ignored. Thus, the equal-thicknessinterference fringes become visible and the homogeneity of the surfacedistribution of the output luminance of the optical conductor degrades.Thus, according to the third aspect of the present invention, theoccupying probability P is selected so that the relation of P<Pc issatisfied.

In addition, as the lower limit of the occupying probability P, a finitevalue that is not zero is required. To accomplish the function asspacers on the entire surface of the lens array sheet, although theoccupying probability P varies corresponding to the size of the clustermember elements, the relation of P≧0.2 should be satisfied.

In reality, the value of the occupying probability P is designatedcorresponding to the size of the cluster member elements (normally, thelength of the diagonal line on the rear surface; or the width a and thedepth b when the cluster member elements are formed in a rectangularparallelepiped shape), the lattice constant, and the bendingcharacteristic of the lens array sheet. FIGS. 16 to 20 show examples ofcomputer simulations using uniform random numbers in the case that thelattice constant of a square lattice is 100 μm and the occupyingprobability P is 0.2, 0.3, 0.4, 0.5, and 0.6. For simplicity, in FIGS.16 to 20, only a 25×25 lattice points of the entire lattice are shown.In the range of 0.2≦P≦0.5, the maximum value Rs^(max) of the averagerotating radius Rs is in the range from 110 μm to 630 μm. The area ratioSr of the cluster members is in the range from 18 to 54%. Thus, Rs^(max)is in the preferable range. In addition, Sr is also in a preferablerange that will be described later.

FIGS. 13A, 13B, and 13C show examples for fusing cluster memberelements. FIG. 13A shows the case that the rear surface allocated toeach lattice point 8 (namely, each structural element) selected with apredetermined occupying probability P is formed in a square shape andcluster member elements 24 are adjacently disposed, each of which issmaller than the length of a lattice element of a square lattice. FIG.13B shows the case that the shape of a connecting portion 26 of theadjacent cluster member elements is smooth and linear. The clustermember elements in FIG. 12B are fused by the method shown in FIG. 13B.FIG. 13C shows the case that the width of the connecting portion 26 issmaller than that of each of the cluster member elements. In this case,even if the cluster member elements are successively fused, the sidesurface of the cluster member elements is prevented from becominglinearly long. Thus, the moire fringes can be more prevented than thecase shown in FIG. 13B. However, in the case shown in FIG. 13C, sincethe shape is complicated, higher machining accuracy is required than thecase shown in FIG. 13B.

The two-dimensional distribution of the cluster members 21 on the lensarray sheet is preferably a random distribution. In the third aspect ofthe present invention, the cluster members 21 have randomnesscorresponding to the theory of percolation (it is clear from thedistribution in the case that the center of gravity of an area elementon the rear surface of each cluster member is treated as arepresentative value of the coordinates). If cluster members areperiodically formed, since they periodically overlap with lens elementson the opposite surface of the lens array sheet (in most cases, the lenselements are periodically formed), the moire fringes take place. Inaddition, when the lens array sheet is used for a back light of a colorliquid crystal display device, the cluster members interfere with theplacement period of pixels. of the display device. Thus, the moirefringes tend to take place. To solve such a problem, when the clustermembers are randomly formed, the occurrence of the moire fringes can beprevented.

However, even if the cluster members 21 are randomly formed, when theshape of each cluster member element that composes a cluster member isthe same and the orientation thereof is the same, since the sidesurfaces of the cluster members composed of the side surfaces of thecluster member elements are oriented in the same direction, a set ofsmall side surfaces in the same direction virtually forms a large sidesurface. Even if the cluster members are randomly formed, when aperiodical lattice (for example, a square lattice) is used as atwo-dimensional lattice and the shape and size of each of cluster memberelements disposed at each lattice point are the same, periodicity takesplace. The virtual side surface interferes with the surface having thelens elements that compose the lens array and thereby the moire fringesmay take place. Thus, it is preferable to designate a particularrelation between the surface that composes the lens elements and theside surface that has the cluster members.

FIGS. 11A and 11B are schematic diagrams for explaining a structure forpreventing the moire fringes from taking place. For example, as shown inFIG. 11A, assume a structure of which the lens array of the lens arraysheet 1 is composed of triangular prism lenses as lens elements 41. Thelight emitting surface of the lens array sheet 1 is in parallel with theX-Y plane. The light emitting surface is referred to as a horizontalsurface. The normal direction perpendicular to the light emittingsurface is the direction of the axis Z (not shown). The structuralsurfaces of each lens element 41 are inclined surfaces 42 that form thetop and the bottom of a triangular prism. The line of intersection ofthe inclined surfaces and the horizontal surface is in parallel with theaxis X (in this case, the coordinates are defined so that the axis X isin parallel with the line of intersection). Strictly speaking, theinclined surface is a finite surface. The horizontal surface can bedefined in various manners depending on the coordinates of the axis Z.The inclined surface does not intersect with the horizontal surfacedepending on a condition. In this example, the line of intersectionrepresents the line of which the inclined surface is extended andintersected with the horizontal surface. When triangular prisms as lenselements are one-dimensionally arrayed, there is one line ofintersection. On the other hand, when quadrangular prisms as lenselements are two-dimensionally arrayed, there may be two or more linesof intersection. In this case, the lines of intersection may be notperpendicular to each other.

FIG. 11B is a schematic diagram showing the case that X-Y coordinatescorresponding to the line of intersection of the lens elements 41 of thetriangular prisms is overlaid with X'-Y' coordinates corresponding toaxis X' of one line of intersection obtained from the cluster 2.

The orientations of the cluster members 21 are arranged. There are twolines of intersection of side surfaces of the cluster members 21 and thehorizontal surface of the lens array sheet. The two lines ofintersection are perpendicular to each other. They are lines ofintersection in parallel with the axis X' and the axis Y'. The axis X'and the axis X form an angle α.

When the angle α between the axis X and the axis X' is zero, the axis Xis in parallel with the axis X'. Thus, the moire fringes tend to takeplace. However, when the line of intersection of each lens element hasan angle of 5° to the line of intersection of each cluster member, themoire fringes can be prevented. In other words, in the case that thelines of intersection of the cluster members define an orthogonalcoordinate system, when the angle α is in the range from 5° to 85° (inthe clockwise direction), more preferably in the range from 10° to 80°(in the clockwise direction), the moire fringes can be effectivelyprevented. In addition, the angle α is preferably in the range from -5°to -85° (in the counterclockwise direction), more preferably in therange from -10° to -80° (in the counterclockwise direction). In the caseof the rectangular parallelepiped shape, when the angle α exceeds 85°,the angle to the line of intersection of the side surface becomes large.Thus, since the relation between the adjacent side surfaces (90° to theside surface) becomes almost parallel. Consequently, due to the relationwith the adjacent side surfaces, the moire fringes tend to take place.When the side surface of the cluster members composed of cluster memberelements with a rear surface formed in a rectangular shape that definesthe orthogonal coordinate system has an angle exceeding 5° to thehorizontal direction, the moire fringes can be prevented.

When the cluster members are composed of for example rectangularparallelepiped members and the angle between the line of intersection ofa particular side surface of each rectangular parallelepiped member andthe horizontal surface of the lens array sheet and the line ofintersection of the surface of each lens element and the horizontal lineexceeds 5°, it is not necessary to arrange the orientations of all thecluster members (formed in the rectangular parallelepiped shape)allocated to structural elements. For example, even if 1% of all clustermembers are arranged in parallel, when they are not disposed at adjacentlattice elements, the parallel relation of which the moire fringes takeplace is not defined.

Thus, in claim 18 of the third aspect of the present invention, "eachrectangular parallelepiped member" where the line of intersection of aside surface of each rectangular parallelepiped member is not inparallel with the line of intersection of a lens element does not meanthat all rectangular parallelepiped members that are formed do not havea non-parallel relation, but that even if part of rectangularparallelepiped members have a parallel relation, the non-parallelrelation takes place as the general situation.

As the shape of each cluster member according to the third aspect of thepresent invention, a prism shape can be used instead of the rectangularparallelepiped shape as described above. When the above-describedrectangular parallelepiped members are disposed to structural elementsin the same direction, the angle of each adjacent side is 90°. Thus,whenever the rectangular parallelepiped members are rotated for 90°, thesame situation takes place. However, in the case of the rectangularparallelepiped shape, since each opposite side surfaces are in parallel,it is necessary to consider two lines of intersection that areperpendicular to each other. However, in the case of the prism shapeother than the rectangular parallelepiped shape, for example atriangular prism shape, the number of lines of intersection to beconsidered is three. In the case of a pentagonal prism shape, the numberof lines of intersection to be considered is five. In these cases, thenumber of lines of intersection to be considered is greater than that inthe case of the rectangular parallelepiped shape. Thus, the probabilityof the occurrence of the moire fringes increases. Consequently, thedegree of freedom of designing the lens array sheet decreases. Even inthe case of a free quadrilateral shape where each adjacent side does notform a right angle, the number of lines of intersection to be consideredis as many as four. Thus, even if a quadrangular prism shape with a rearsurface that has a parallelogram shape or a rhombus shape is used, aswith the case of the rectangular parallelepiped shape, the occurrence ofthe moire fringes can be prevented. However, the cluster members in therectangular parallelepiped shape is more easily fabricated than those inthe quadrangular prism shape with the rear surface having aparallelogram shape or a rhombus shape.

In the case that the lines of intersection of side surfaces are notstraight lines, there is an n-side prism (where n is infinite) (namely,a circular cylinder shape or an elliptic cylinder shape where the sidesurface is a curved surface). In this case, when an original press filmfor forming cluster members and thereby the cluster member elements isproduced by a horizontal scanning method using a scanner or the like,since the cluster members are very small, the contour of for example acircular shape of side surfaces that are not in parallel with orperpendicular to scanning lines are rugged. Thus, a smooth side surfaceof the cylinder cannot be obtained.

The moire fringes that take place in the relation between the structuralsurface of each cluster member and the structural surface of each lenselement. In other words, when the cluster members are allocated tostructural elements in the same orientation, the side surfaces thereofare arranged. Thus, a line of intersection that can be recognized isdefined. This is because the relation between the line of intersectionof each cluster member and the line of intersection of each lens elementtakes place. However, even if the shapes of the cluster members are thesame, when they are randomly formed (namely, the cluster members arerotated around the axis Z that is perpendicular to the X-Y plane), theline of intersection of a side surface of each cluster member has anangle that is dispersed. Thus, there is no line of intersection definedat a predetermined angle. In such a manner, the occurrence of the moirefringes can be prevented. In this point, the circular cylinder shape,the elliptic cylinder shape, and the like are superior to the othershapes. However, as described above, the side surface that is a smoothlycurved surface is difficult to fabricate.

The distribution density of the cluster members is designated so thatthe lens array sheet is not bent and thereby the equal-thicknessinterference fringes do not take place. In addition, even if the lensarray sheet has a proper rigidity, a homogeneous distance between thelens array sheet and the optical conductor or between the lens arraysheets can be maintained so that a small difference of the distancesprevents the equal-thickness interference fringes from taking place.Thus, the lattice size of the tow-dimensional lattice can be designatedcorresponding to the distribution density.

In the case that two lens array sheets are layered, the distributiondensity of which the sectional area of each cluster member is zero(namely, the distribution density of the cluster members) is preferablydesignated to the relation of t<2p (where t is the average distance ofadjacent cluster members formed on the rear surface of the upper lensarray sheet; and p is the repetitive period of the lens elements formedon the front surface of the lower lens array sheet). Thus, sincesupporting contacts between the cluster members 21 formed on the rearsurface of the upper lens array sheet and the lens elements 41 formed onthe front surface of the lower lens array sheet are prevented from beingbent regardless of the sectional area of the cluster member elements,the distance between the upper and lower lens array sheets does notbecome heterogeneous. Consequently, the equal-thickness interferencefringes do not take place. In addition, the distance between the upperand lower lens array sheets can be prevented from becoming smaller thanthe wave length of the source light. The average distance t is morepreferably in the range of t<0.5 p. However, in reality, since thecluster member elements have a finite sectional area and they areconnected, even if t is greater than 0.5 p, this effect can besatisfactorily accomplished.

On the other hand, as a distribution density for preventing theequal-thickness interference fringes from taking place even if the lensarray sheet bends in the case that the sectional area of each clustermember is finite, the area ratio Sr (=(Sp/St)×100) of the sum of Sp ofthe sectional areas of the cluster members against the entire area St ofwhich the lens array sheet 1 faces the optical conductor 51 ispreferably in the range from around 0.01 to 60%. As the function ofspacers, the number of cluster members should be as small as possible.However, to prevent the lens array sheet from bending, a proper numberof cluster members are required. When the lens array sheet is used as asurface light source along with an optical conductor (that will bedescribed later), a proper number of cluster members are required tohomogenize the surface distribution of the luminance. In particular,when the lens array sheet has a bending characteristic equivalent to abiaxial drawing polyethylene terephtalate with a thickness of 50 to 100μm, Sr is preferably in the range from 20 to 60%.

The above-described randomizing process corresponding to the theory ofpercolation can be performed by for example a computer. The calculatedresult is printed on an original press film. Alternatively, a block ofcluster members that have been randomized can be repeatedly formed invertical and horizontal directions so as to form cluster members with arequired area.

In the third aspect of the present invention, when the above-describedcluster members are formed on one surface of the lens array sheet, lightthat is emitted out of the angular range of visual field is notincreased and thereby the luminance is not decreased. In addition, theequal-thickness interference fringes and the moire fringes can beprevented. Thus, the lens array sheet can homogeneously distribute lighton the entire surface of the optical conductor with a homogeneoussurface distribution.

Next, a surface light source and a transmission type display devicehaving the lens array sheet according to the first to third aspects ofthe present invention will be described.

As shown in FIGS. 23A and 23B, the lens array sheet 1 according to thefirst to third aspects of the present invention may be composed of threelayers that are a flat transparent substrate 3, a cluster 2, and a lensarray 4. In this case, the cluster 2 is composed of a large number ofcluster members 21. The cluster 2 is formed on one surface of thetransparent substrate 3. The lens array 4 is formed on the other surfaceof the transparent substrate 3. Alternatively, as shown in FIG. 22, thelens array 4 and the transparent substrate 3 may be integrally formed.On the resultant structure, the cluster 2 may be layered so as toaccomplish a two-layer type lens array sheet. As another alternativestructure, as shown in FIG. 21, the lens array 4, the transparentsubstrate 3, and the cluster 2 may be integrally formed so as toaccomplish a one-layer type lens array sheet. In such an integrallyformed lens array sheet, the transparent substrate is not alwaysrequired. In this case, the cluster may be formed on the rear surface ofthe lens array. The integral member of the lens array and the clustercan be treated as the transparent substrate.

In FIG. 23B, a transparent cluster base 32 and the cluster 2 areintegrally formed. The cluster base 32 is formed on one entire surfaceof the transparent substrate 3. In this structure, the cluster base 32may be is formed on the transparent substrate 3 along with the cluster.In this case, the cluster base 32 can be considered as a part of thetransparent substrate 3.

The transparent substrate, the cluster, and the lens array are composedof a transparent material. Depending on the application, such a materialmay be colored or semitransparent. In addition, since the size ofcluster members is small, they may be transparent as long as they areinvisible.

As examples of the transparent material for the transparent substrate,the lens array, and the cluster, a polyester resin (such as polyethyleneterephtalate or polybutylene terephtalate), an acrylic resin (such aspolymethyl methacrylate), a thermoplastic rein (such as polycarbonateresin, polyethylene resin, or polymethyl pentene), or anionizing-radiation-curable resin (such as polyester acrylate, urethaneacrylate, or epoxy acrylate that is composed of a monomer or the likesuch as oligomer and/or acrylate). The ionizing-radiation-curable resinis hardened with ionizing radiation such as ultraviolet ray orradiation. The refractive index of such a resin is normally in the rangefrom 1.49 to 1.55. As another material other than such a resin, glass,ceramics, or the like can be used as long as it has a good transparentcharacteristic.

The total thickness of the lens array sheet is normally in the rangefrom 20 to 1000 μm.

As an example of the lens array of the lens array sheet according to thefirst to third aspects of the present invention, as shown in FIG. 24, alinear array of prism lens of which lens elements 41 formed in atriangular prism shape are adjacently arrayed so that the longer axis(edge line) of each lens element 41 is one-dimensionally arranged inparallel (linear array). This lens array is referred to a lenticularlens array in a wide sense. Alternatively, as shown in FIG. 27, lenselements 41 formed in a semi-sphere shape are two-dimensionally arrayedas a fly-eye lens array.

Examples of the sectional shape of the lens elements, as shown in FIGS.25 and 26, are smoothly continuous curves such as a circle shape, anellipse shape, a cardioid shape, a Rankine's egg shape, a cycloid shape,and an involute curve shape. Alternatively, as shown in FIG. 24, as thesectional shape of the lens elements, part or all of a polygon shapesuch as a triangle shape, a quadrilateral shape, or a hexagon shape canbe used.

In addition, as lens elements that are two-dimensionally arrayed,pyramid lenses can be used.

The lens elements may be formed in a convex shape as shown in FIGS. 24,25, 27, and 28 or a concave shape as shown in FIG. 26. Among theseshapes of the lens elements, a circular cylinder shape or an ellipticcylinder shape are preferable from view points of easy design andfabrication, light condensing characteristic, and light diffusioncharacteristics (low half value angle and low side robe light (thatlargely deviates from the normal of the light emitting surface of thelens array sheet), isotropy of luminance in the half value angle, andluminance in normal direction). In particular, the elliptic cylindershape of which the normal direction of the surface light source accordswith the longer diameter is preferable for high luminance.

In FIGS. 24, 25, 26, 27, and 28 for explaining the shapes of the lenselements, the clusters are omitted.

As a fabrication method for fabricating a lens array sheet, toaccomplish a one-layer type lens array sheet shown in FIG. 21, diescorresponding to the lens array 4 and the cluster 2 can be used by aknown heat press method or a known injection casting method using athermoplastic resin as disclosed in for example Japanese PatentLaid-Open Publication No. 56-157310. Alternatively, the lens array sheetmay be fabricated by an injection casting method using a radiationcurable resin or a thermosetting resin.

As another fabrication method as disclosed in for example JapanesePatent Laid-Open Publication No. 5-169015, a form plate cylinder(cylindrical casting mold) having a concave portion (precisely speaking,a cluster shape) corresponding to the shape of a desired lens array isfilled with an ionizing radiation curable resin solution. A transparentsubstrate sheet is layered on the resultant structure. An ionizingradiation such as an ultraviolet ray or an electron beam is applied tothe transparent sheet side (or to the form plate cylinder when it iscomposed of glass or the like that is transparent). Thus, the ionizingradiation curable resin solution is cured (cross linked or polymerizedto solid). Thereafter, the transparent substrate sheet is peeled offfrom the form plate cylinder along with the hardened resin. Thus, anintermediate sheet of the lens array sheet of which the lens array 4 inthe desired shape is formed on the transparent substrate sheet isobtained.

Thereafter, the similar process is performed on the rear surface of theintermediate sheet with a form plate cylinder corresponding to the shapeof a desired cluster. Thus, the lens array sheet having the cluster andthe lens array according to the first to third aspects of the presentinvention can be accomplished.

It is possible to form a cluster before a lens array.

FIG. 33 is a sectional view showing a fabrication apparatus forfabricating a lens array sheet using such an ionizing radiation curableresin.

In the fabrication apparatus shown in FIG. 33, reference numeral 71 is aform plate cylinder having a concave portion 72 corresponding to acluster 2 (or a lens array 4). In the drawing, concave portions arerepresented by square sections. The form plate cylinder is rotatedaround the axis thereof in the arrow direction. Reference numeral 73 isan ionizing radiation curable resin solution filled in the concaveportions. Reference numeral 3 is a sheet type transparent material(substrate). Reference numeral 74 is a pressing roll that contacts theform plate cylinder so as to press the transparent substrate 3 to theform plate cylinder 71. Reference numeral 75 is a guide roll thatsupports the traveling of the transparent substrate 3. Reference numeral76 is a peeling roll. Reference numerals 77a and 77b are ionizingradiation radiating units that harden ionizing radiation curable resinsolution to the solid state. Reference numeral 21 is a cluster formed onthe transparent substrate 3 as a hardened substance of the ionizingradiation curable resin solution. Reference numeral 11 is anintermediate sheet having the cluster 21 (or the lens array 4) formed onthe transparent substrate 3. Reference numeral 78 is a coating unit thatcoats the ionizing radiation curable resin solution.

In the above-described fabrication method, as the sheet type transparentsubstrate, a polyester resin such as polyethylene terephtalate orpolybutylene terephtalate can be used. The thickness of the transparentsubstrate is normally in the range from 10 to 1000 μm although itdepends on the workability of the apparatus and so forth.

In such a method, as shown in FIG. 23, a three-layer type lens arraysheet having layers of a cluster 2, a lens array 4, and a transparentsubstrate 3 can be accomplished.

In a two-layer type lens array sheet as shown in FIG. 22, anintermediate sheet having a lens array is fabricated by theabove-described rotary casting method, injection casting method, or thelike. Thereafter, by the method using the above-described ionizingradiation curable resin and the form plate cylinder or a flat castingmold, a cluster is formed.

Next, an edge light type surface light source 100 according to the firstto third aspects of the present invention can be accomplished bydisposing the lens array sheet according to the first to third aspectsof the present invention on the light emitting surface of a surfacelight source composed of a light source, an optical conductor, areflection layer, and so forth that are known.

FIG. 30 is a perspective view showing the edge light type surface lightsource according to an embodiment of the first to third aspects of thepresent invention. The edge light type surface light source comprises anoptical conductor (light guide) 51, a linear type or point type lightsource 52, a light reflection layer 53, and a lens array sheet 1. Thelight source 52 is adjacently disposed at least one of the side edgesurfaces. The light reflection layer 53 is formed on the rear surface ofthe light conductor 51. The lens array sheet 1 accords with the presentinvention. The edge light type surface light source further comprises alamp house 54 having an inner reflection surface disposed around thelight source 52.

The optical conductor 51 is a transparent plate composed of an acrylicresin, a polycarbonate resin, or the like with a thickness of 1 to 10mm. The light source 52 is a linear light source such as an air-coolingcathode ray tube. The light reflection layer 53 is formed by coating awhite paint on a base material or depositing or plating a metal film ona metal film that has been sand-blasted so as to diffuse and reflectlight. In addition, a white light diffusion dot pattern may be formed soas to homogenize the amount of light emitted from the light emittingsurface.

Moreover, a light diffusion sheet may be disposed between the opticalconductor 51 and the lens array sheet 1 so as to homogeneously diffuselight and allow the light diffused dot pattern on the light reflectionlayer 53 to become invisible.

When the lens array sheet is disposed at the surface light source, toprovide light diffusion angles in two directions (vertical andhorizontal directions), as shown in FIG. 29, two lens array sheets maybe layered so that the edge lines of the lens elements of the first lensarray sheet are perpendicular to those of the second lens array sheet.In this case, when the orientation of the surface on which the lensarray is formed (hereinafter referred to as a lens surface) of the firstlens array sheet is the same as that of the second lens array sheet,transmissivity of light is improved and the moire fringes between thelens surface of the lower lens array sheet and the cluster of the rearsurface of the upper lens array sheet can be prevented. However, it ispossible to dispose the two lens array sheets so that the lens surfacesthereof face each other.

In the case that two lens array sheets are used, when the lightdiffusion sheet 56 that has two surfaces with concave and convexportions (projection) whose height is homogeneous to or greater than thewave length of the light source is disposed between a lens array sheetand an optical conductor 51, the cluster on the rear surface of thelower lens array sheet can be omitted. This is because the concave andconvex portions on the top and rear surfaces of the light diffusionsheet prevent the optical contact from taking place. In this structure,although two lens array sheets are used, one lens array sheet is aconventional lens array sheet 19 whose rear surface is flat. Thus, thefabrication cost can be reduced. With the light diffusion sheet, lightcan be homogeneously diffused. Although the amount of light that isemitted out of the angular range of visual field increases, the lightdiffusion dot pattern of the light reflection layer on the rear surfaceof the optical conductor can become invisible.

FIG. 32 shows such an edge light type surface light source 101.

The bottom-flat lens array sheet has an lens array of which lenselements are one- or two-dimensionally arrayed on the front surface ofthe transparent substrate. Thus, the rear surface of the lens arraysheet is flat. Consequently, the structure of the lens array sheet isthe same as that of the lens array sheet 1 according to the first tothird aspects of the present invention except that the cluster 2 on therear surface is omitted. Thus, the material and fabrication method arethe same as those of the lens array sheet according to the first tothird aspects of the present invention.

The light diffusion sheet has concave and convex portions on the top andrear surfaces thereof so as to prevent the optical conductor fromcontacting the rear surface of the lower lens array sheet. The lightdiffusion is performed by the concave and convex portions or a lightdiffusion agent in the light diffusion sheet. As the light diffusionsheet, a known light diffusion sheet can be used. For example, the lightdiffusion sheet can be composed by dispersing light diffusion particlesto a transparent substrate such as an acrylic resin or by forming anemboss pattern such as a sand-face pattern on the front surface of thetransparent resin.

When the edge light type surface light source according to the first tothird aspects of the present invention is disposed on the rear surfaceof the light transmitting display device such as a transmission typeliquid crystal display device or an advertisement board. FIG. 31 is atransmission type display device 200 of which the transmission typedisplay device 6 is disposed to the surface light source 100 (shown inFIG. 30) according to the first to third aspects of the presentinvention.

Examples Corresponding to First Aspect of Present Invention

<First Example>

By a fabrication apparatus as shown in FIG. 33, an intermediate sheethaving a lens array composed of lens elements formed in an isoscelestriangle shape whose apex angle β (see FIG. 24) was 100° was formed bycoating an ultraviolet ray curable resin solution mainly composed ofurethane acrylate prepolymer on a biaxial drawing polyethyleneterephtalate with a thickness of 100 μm as a transparent substrate.

Likewise, by the apparatus shown in FIG. 33, the same ultraviolet raycurable resin solution was coated on the rear surface of theintermediate sheet so as to randomly form a large number of clustermembers in a rectangular parallelepiped shape of which the height H was10 μm, the width a was 125 μm, and the depth b was 125 μm. Thus, a lensarray sheet according to the first aspect of the present invention wasobtained.

The form plate cylinder (shown in FIG. 33) for randomly forming thecluster members was produced in the following manner. Image data wasgenerated by a computer image process. As shown in FIGS. 5 and 6,cluster members are represented in black and overlap portions arerepresented in white (concentration=zero). The image data was exposed toa reprophotographic silver-salt-photosensitive film by using areprophotographic scanner. Thus, a printing film was obtained(rectangles representing cluster members are black, whereas the overlapportions and non-rectangular portions are transparent). A photosensitiveresist was coated on the front surface of a copper cylinder. Theprinting film was contacted to the resultant cylinder and then exposed.The resultant cylinder was developed so as to remove the resist of thenon-sensitive portion (inside of the rectangles 21 of FIG. 5B). Thus, aresist pattern of which a hardened film was formed on the exposedportion was formed. The resist was etched out with a ferric chloridesolution from the front surface of the cylinder. Thus, a form platecylinder having rectangular parallelepiped shape concave portions thatwere randomly distributed was obtained. The depth of all the rectangularparallelepiped shape portions of the press was the same.

The orientation of each of the rectangular parallelepiped shape clustermembers was the same. The line of intersection of the side surface ofeach rectangular parallelepiped shape cluster member and the horizontalsurface of the lens array sheet (the axis X' of FIG. 4) was in parallelwith the line of intersection of the structural surface of each lenselement and the horizontal surface of the lens array sheet (the axis Xof FIG. 4). The ratio Sr of the cluster members to the entire sectionalarea was 60 %!.

<Second Example>

The lens array sheet according to the first aspect of the presentinvention was obtained in the same manner as the first example exceptthat the height H of each rectangular parallelepiped cluster member was15 μm.

<Third Example>

The lens array sheet according to the first aspect of the presentinvention was obtained in the same manner as the first example exceptthat the height H of each rectangular parallelepiped cluster member was20 μm.

<Fourth Example>

The lens array sheet according to the first aspect of the presentinvention was obtained in the same manner as the first example exceptthat the rectangular parallelepiped cluster members were rotated, thatthe orientation thereof was the same, and that the angle between theaxis X' and the axis X of FIG. 4 was 10°.

<First Compared Example>

A lens array sheet was obtained in the same manner as the first exampleexcept that the lens array was formed on the front surface of theintermediate sheet and that the cluster was not formed on the rearsurface thereof.

<Second Compared Example>

A matted lens array sheet was obtained in the same manner as the firstexample except that after the intermediate sheet having the lens arrayon the front surface, a coating solution of which acrylic resin beadswith particle diameters of 2 to 20 μm of 3 weight % was added to a twopart curable type urethane resin was coated on the rear surface of theintermediate sheet for 5 g/m² by gravure coating method.

<Third Compared Example>

A rear-surface-matted lens array sheet was obtained in the same manneras the first example except that the form plate cylinder was composed bysand-blasting sand in #80 sphere shape on the front surface of thecylinder.

Table 1 shows the characteristics of the above-described examples andcompared examples for the following evaluation items.

(a) Contact resistance: When the rear surface of the lens array sheetdid not optically contact the other plate (the light emitting surface ofthe optical conductor), this item was evaluated as "O". Otherwise, thisitem was evaluated as "X". In other words, when the region in thevicinity of the light source of the light emitting surface was brighterthan other portions, this item was evaluated as "X". When a brighterregion in the vicinity of the light source was not viewed, this item wasevaluated as "O".

(b) Equal-thickness interference fringes: When the lens array sheet 1and another flat plate were layered and observed, if equal-thicknessinterference fringes were not viewed, this item was evaluated as "O".Otherwise, this item was evaluated as "X".

(c) Moire fringes: When the lens array sheet was observed, if moirefringes were not viewed between the cluster and the lens array, thisitem was evaluated as "O". Otherwise, this item was evaluated as "X".

(d) Luminance factor: The portion other than the lens array sheet wastreated as the same edge light type surface light source. The lens arraysheet of each of the above-described examples and compared examples wasdisposed on the front surface of the optical conductor so that the lensarray faced the output side of the surface light source. The luminancewas measured by a variable angle photometer GONIOPTOMETER made byMURAKAMI SHIKISAI KENKYUJYO K.K. With a reference of the measurement,the luminance in the normal direction of the first compared example ofwhich no cluster was not formed on the rear surface was designated to100%.

(e) Angular range of visual field: Assuming that the luminance in thenormal direction was 100%, the angular range where the luminance became50% or greater than 50% to the normal (half value angle) was measured inthe same manner as the item (d).

                  TABLE 1                                                         ______________________________________                                        Comparison of Characteristics of Examples and                                 Compared Examples                                                                               Equal-                 Angular                                                thickness              range of                                      Contact  interferen                                                                             Moire Luminance                                                                             visual                                        resistance                                                                             ce fringes                                                                             fringes                                                                             factor  %!                                                                            field  0!                            ______________________________________                                        First example                                                                          ∘                                                                          ∘                                                                          X     98      75                                   Second   ∘                                                                          ∘                                                                          X     97      75                                   example                                                                       Third example                                                                          ∘                                                                          ∘                                                                          X     94      75                                   Fourth example                                                                         ∘                                                                          ∘                                                                          ∘                                                                       98      75                                   First compared                                                                         X        X        ∘                                                                       100     *                                    example                                                                       Second   ∘                                                                          X        ∘                                                                       92      80                                   compared                                                                      example                                                                       Third    ∘                                                                          ∘                                                                          ∘                                                                       83      82                                   compared                                                                      example                                                                       ______________________________________                                         *Not measurable. At a position with a distance of 2 cm to the light           source, the emitted light remarkably becomes dirk.                       

Examples Corresponding to Second Aspect of Present Invention

<First Example>

By a fabrication apparatus as shown in FIG. 33, an intermediate sheethaving a lens array composed of lens elements formed in an isoscelestriangle shape whose apex angle β (see FIG. 24) was 100° was formed bycoating an ultraviolet ray curable resin solution mainly composed ofurethane acrylate prepolymer on a biaxial drawing polyethyleneterephtalate with a thickness of 100 μm as a transparent substrate.

Likewise, by the apparatus shown in FIG. 33, the same ultraviolet raycurable resin solution was coated on the rear surface of theintermediate sheet so as to randomly form a large number of clustermembers in a rectangular parallelepiped shape of which the height H was10 μm, the width a was 100 μm, and the depth b was 100 μm. Thus, a lensarray sheet according to the second aspect of the present invention wasobtained.

The form plate cylinder 72 (shown in FIG. 33) for randomly forming thecluster members was produced in the following manner. Image data wasgenerated by a computer image process. As a two-dimensional lattice, asquare lattice as shown in FIG. 7A (a=b=250 μm) was used. Coordinates ofeach lattice point were varied in the directions of the X and Y axeswith random numbers (in the range from -0.5 to +0.5) with α=β350 μm inthe Formula 3.2! and Formula 3.3!. As shown in FIG. 6, cluster membersare represented in black and overlap portions are represented in white(concentration=zero). The image data was exposed to a reprophotographicsilver-salt-photosensitive film by using a reprophotographic scanner.Thus, a printing film was obtained (in FIG. 5C, rectangles representingcluster members are black, whereas the overlap portions andnon-rectangular portions are transparent). A photosensitive resist wascoated on the front surface of a copper cylinder. The printing film wascontacted to the resultant cylinder and then exposed. The resultantcylinder was developed so as to remove the resist of the non-sensitiveportion (inside of the rectangles). Thus, a resist pattern of which ahardened film was formed on the exposed portion was formed. The resistwas etched out with a ferric chloride solution from the front surface ofthe cylinder. Thus, a form plate cylinder having rectangularparallelepiped shape concave portions that were randomly distributed wasobtained. The depth of all the rectangular parallelepiped shape portionsof the press was the same.

The orientation of each of the rectangular parallelepiped shape clustermembers was the same. The line of intersection of the side surface ofeach rectangular parallelepiped shape cluster member and the horizontalsurface of the lens array sheet (the axis X' of FIG. 4) was in parallelwith the line of intersection of the structural surface of each lenselement and the horizontal surface of the lens array sheet (the axis Xof FIG. 4). The ratio Sr of the cluster members to the entire sectionalarea was 15 %!.

<Second Example>

The lens array sheet according to the second aspect of the presentinvention was obtained in the same manner as the first example exceptthat the height H of each rectangular parallelepiped cluster member was15 μm.

<Third Example>

The lens array sheet according to the second aspect of the presentinvention was obtained in the same manner as the first example exceptthat the height H of each rectangular parallelepiped cluster member was20 μm.

<Fourth Example>

The lens array sheet according to the second aspect of the presentinvention was obtained in the same manner as the first example exceptthat the rectangular parallelepiped cluster members were rotated, thatthe orientation thereof was the same, and that the angle between theaxis X' and the axis X of FIG. 4 was 10°.

<First Compared Example>

A lens array sheet was obtained in the same manner as the first exampleexcept that the lens array was formed on the front surface of theintermediate sheet and that the cluster was not formed on the rearsurface thereof.

<Second Compared Example>

A matted lens array sheet was obtained in the same manner as the firstexample except that after the intermediate sheet having the lens arrayon the front surface, a coating solution of which acrylic resin beadswith particle diameters of 2 to 20 μm of 3 weight % was added to a twopart curable type urethane resin was coated on the rear surface of theintermediate sheet for 5 g/m² by gravure coating method.

<Third Compared Example>

A rear-surface-matted lens array sheet was obtained in the same manneras the first example except that the form plate cylinder was composed bysand-blasting sand in #80 sphere shape on the front surface of thecylinder.

Table 2 shows the characteristics of the above-described examples andcompared examples.

The evaluation criteria of the second aspect of the present inventionare the same as those of the first aspect of the present invention.

                  TABLE 2                                                         ______________________________________                                        Comparison of Characteristics of Examples and                                 Compared Examples                                                                              Equal-                  Angular                                       Contact thickness       Luminance                                                                             range of                                      resist- interference                                                                            Moire factor  visual                                        ance    fringes   fringes                                                                              %!     field  o!                            ______________________________________                                        First example                                                                          ∘                                                                         ∘                                                                           X     98      75                                   Second   ∘                                                                         ∘                                                                           X     97      75                                   example                                                                       Third example                                                                          ∘                                                                         ∘                                                                           X     94      75                                   Fourth example                                                                         ∘                                                                         ∘                                                                           ∘                                                                       98      75                                   First compared                                                                         X       X         ∘                                                                       100     *                                    example                                                                       Second   ∘                                                                         X         ∘                                                                       92      80                                   compared                                                                      example                                                                       Third    ∘                                                                         ∘                                                                           ∘                                                                       83      82                                   compared                                                                      example                                                                       ______________________________________                                         *Not measurable. At a position with a distance of 2 cm to the light           source, the emitted light remarkably becomes dirk.                       

Example Corresponding to Third Aspect of Present Invention

<First Example>

By a fabrication apparatus as shown in FIG. 33, an intermediate sheethaving a lens array composed of lens elements formed in an isoscelestriangle shape whose apex angle β (see FIG. 24) was 100° was formed bycoating an ultraviolet ray curable resin solution mainly composed ofurethane acrylate prepolymer on a biaxial oriented polyethyleneterephtalate with a thickness of 100 μm as a transparent substrate.

Likewise, by the apparatus shown in FIG. 33, the same ultraviolet raycurable resin solution was coated on the rear surface of theintermediate sheet so as to randomly form a large number of clustermembers in a rectangular parallelepiped shape of which the height H was10 μm, the width a was 100 μm, and the depth b was 100 μm. The clustermember elements were disposed in a two-dimensional square lattice with4096 (long)×4096 (wide) lattice points with a lattice constant of 100 μmas a site percolation model of which the percolation probability P is0.2. Since one side of the bottom side of each cluster member elementwas homogeneous to the lattice constant, adjacent cluster memberelements were directly connected so as to form cluster members. Thus, acluster as shown in FIG. 16 was formed. Consequently, the lens arraysheet according to the third aspect of the present invention wasobtained.

The form plate cylinder (shown in FIG. 33) for forming the clustermembers had a front surface corresponding to the reverse shape of thecluster. Image data was generated by a computer image process usingMonte Carlo simulation data. The image data was exposed to areprophotographic silver-salt-photosensitive film by using areprophotographic scanner. Thus, a printing film was obtained (in FIG.5C, rectangles representing cluster members are black, whereas theoverlap portions and non-rectangular portions are transparent). Aphotosensitive resist was coated on the front surface of a coppercylinder. The printing film was contacted to the resultant cylinder andthen exposed. The resultant cylinder was developed so as to remove theresist of the non-sensitive portion (inside of the rectangles). Thus, aresist pattern of which a hardened film was formed on the exposedportion was formed. The resist was etched out with a ferric chloridesolution from the front surface of the cylinder. Thus, a form platecylinder having rectangular parallelepiped shape concave portions thatwere randomly distributed was obtained. The depth of all the rectangularparallelepiped shape portions of the form plate was the same.

The orientation of each of the rectangular parallelepiped shape clustermembers was the same. The line of intersection of the side surface ofeach rectangular parallelepiped shape cluster member and the horizontalsurface of the lens array sheet (the axis X' of FIG. 11) was in parallelwith the line of intersection of the structural surface of each lenselement and the horizontal surface of the lens array sheet (the axis Xof FIG. 11). The ratio Sr of the cluster members to the entire sectionalarea was 20 %!. The maximum rotating radius Rs^(max) of the cluster was110 μm.

<Second Example>

The lens array sheet according to the third aspect of the presentinvention was obtained in the same manner as the first example exceptthat the simulation was performed with the occupying probability P=0.3,that Sr=30%, that Rs^(max) =230 μm, and that the cluster pattern was asshown in FIG. 17.

<Third Example>

The lens array sheet according to the third aspect of the presentinvention was obtained in the same manner as the first example exceptthat the simulation was performed with the occupying probability P=0.4,that Sr=40%, that Rs^(max) =260 μm, and that the cluster pattern was asshown in FIG. 18.

<Fourth Example>

The lens array sheet according to the third aspect of the presentinvention was obtained in the same manner as the first example exceptthat the simulation was performed with the occupying probability P=0.5,that SR=50%, that Rs^(max) =630 μm, and that the cluster pattern was asshown in FIG. 19.

<First Compared Example>

A lens array sheet was obtained in the same manner as the first exampleexcept that the lens array was formed on the front surface of theintermediate sheet and that the cluster was not formed on the rearsurface thereof.

<Second Compared Example>

A matted lens array sheet was obtained in the same manner as the firstexample except that after the intermediate sheet having the lens arrayon the front surface, a coating solution of which acrylic resin beadswith particle diameters of 2 to 20 μm of 3 weight % was added to atwo-part curable type urethane resin was coated on the rear surface ofthe intermediate sheet for 5 g/m² by gravure coating method.

<Third Compared Example>

A lens array sheet was obtained in the same manner as the first exampleexcept that the simulation was performed with the occupying probabilityP=0.6 (>Pc), that S_(R) =63 %, that Rs^(max) =size of all lattice(equivalent to infinite cluster), and that the cluster pattern was asshown in FIG. 20.

Table 3 shows the characteristics of the above-described examples andcompared examples.

The evaluation criteria of the third aspect of the present invention arethe same as those of the first aspect of the present invention.

                  TABLE 3                                                         ______________________________________                                        Comparison of Characteristics of Examples and                                 Compared Examples                                                                              Equal-                  Angular                                       Contact thickness               range of                                      resist- interference                                                                            Moire Luminance                                                                             visual                                        ance    fringes   fringes                                                                             factor  %!                                                                            field  o!                            ______________________________________                                        First example                                                                          ∘                                                                         ∘                                                                           Δ                                                                             98      75                                   Second   ∘                                                                         ∘                                                                           Δ                                                                             94      75                                   example                                                                       Third example                                                                          ∘                                                                         ∘                                                                           Δ                                                                             97      75                                   Fourth example                                                                         ∘                                                                         ∘                                                                           Δ                                                                             98      75                                   First compared                                                                         X       X         ∘                                                                       100     *                                    example                                                                       Second   ∘                                                                         X         ∘                                                                       92      80                                   compared                                                                      example  ∘                                                                         Δ   ∘                                                                       99      75                                   ______________________________________                                         *Not measurable. At a position with a distance of 2 cm to the light           source, the emitted light remarkably becomes dirk.                       

Although the present invention has been shown and described with respectto best mode example thereof, it should be understood by those skilledin the art that the foregoing and various others changes, omissions, andaddition in the form and detail thereof may be made therein withoutdeparting from the spirit and scope of the present invention.

What is claimed is:
 1. A lens array sheet, comprising:a transparentsubstrate having a front surface and an opposite rear surface; a lensarray having lens elements arranged on said front surface of thesubstrate; and a cluster of a large number of minute cluster membersprovided on said opposite rear surface of the substrate in a randomdistribution, each of said minute cluster members having a length, awidth, and a height dimension, each of which is in the range of a wavelength of a source light for the lens array sheet to 500 μm.
 2. The lensarray sheet of claim 1,wherein each of said length, width and heightdimensions is up to 125 μm.
 3. The lens array sheet of claim 1,whereineach of said minute cluster members has a rectangular parallelepipedshape with side surfaces.
 4. The lens array sheet of claim 3,whereinsaid lens elements of the lens array form first lines of intersectionwith said front surface, and said side surfaces of the rectangularparallelepiped shape form second lines of intersection with saidopposite rear surface, said first lines of intersection being notparallel to said second lines of intersection.
 5. The lens array sheetof claim 1,wherein said minute cluster members provided on said oppositerear surface have mutually overlapping portions.
 6. The lens array sheetof claim 1,wherein said minute cluster members are arranged in a randomdistribution such that the cluster members are basically on latticepoints of a periodic lattice, respectively, but randomly moved from thelattice points and reallocated.
 7. The lens array sheet of claim1,wherein each of said minute cluster members comprises cluster memberelements each in a prism shape having side surface.
 8. The lens arraysheet of claim 7,wherein each of said cluster member elements has adiagonal line length and a height in the range of a wave length of asource light for the lens array sheet to 500 μm.
 9. The lens array sheetof claim 8,wherein said cluster member elements are disposed to providemutually overlapping structural elements of a percolation cluster formedon lattice points of a two-dimensional lattice, with a percolationprobability being less than the critical percolation concentration Pc.10. The lens array sheet of claim 9,wherein said cluster member elementsare disposed on said lattice points with a percolation probability Pthat is smaller than the critical percolation concentration Pc.
 11. Thelens array sheet of claim 10,wherein said two-dimension lattice is asquare lattice and each of said cluster member elements has arectangular parallelepiped shape.
 12. The lens array sheet of claim1,wherein each of said minute cluster members comprises cluster memberelements, each in a prismoid shape having side surfaces.
 13. The lensarray sheet of claim 12,wherein each of said cluster member elements hasa height as well as a minimum and a maximum diagonal line length at atop and a bottom of the prismoid shape, said height and maximum diagonalline length ranging from a wave length of a source light for the lensarray sheet to 500 μm.
 14. The lens array sheet of claim 13,wherein saidlens elements of the lens array form first lines of intersection withsaid front surface, and said side surfaces of the prismoid shape formsecond lines of intersection with said opposite rear surface, said firstlines of intersection being not parallel to said second lines ofintersection.